Semisimple varieties of implication zroupoids

Abstract

It is a well known fact that Boolean algebras can be defined using only implication and a constant. In fact, in 1934, Bernstein (Trans Am Math Soc 36:876–884, 1934) gave a system of axioms for Boolean algebras in terms of implication only. Though his original axioms were not equational, a quick look at his axioms would reveal that if one adds a constant, then it is not hard to translate his system of axioms into an equational one. Recently, in 2012, the second author of this paper extended this modified Bernstein’s theorem to De Morgan algebras (see Sankappanavar, Sci Math Jpn 75(1):21–50, 2012). Indeed, it is shown in Sankappanavar (Sci Math Jpn 75(1):21–50, 2012) that the varieties of De Morgan algebras, Kleene algebras, and Boolean algebras are term-equivalent, respectively, to the varieties, $\mathbf {DM}$, $\mathbf {KL}$, and $\mathbf {BA}$ whose defining axioms use only the implication $\rightarrow$ and the constant 0. The fact that the identity, herein called (I), occurs as one of the two axioms in the definition of each of the varieties $\mathbf {DM}$, $\mathbf {KL}$ and $\mathbf {BA}$ motivated the second author of this paper to introduce, and investigate, the variety $\mathbf {I}$ of implication zroupoids, generalizing De Morgan algebras. These investigations are continued by the authors of the present paper in Cornejo and Sankappanavar (Implication zroupoids I, 2015), wherein several new subvarieties of $\mathbf {I}$ are introduced and their relationships with each other and with the varieties studied in Sankappanavar (Sci Math Jpn 75(1):21–50, 2012) are explored. The present paper is a continuation of Sankappanavar (Sci Math Jpn 75(1):21–50, 2012) and Cornejo and Sankappanavar (Implication zroupoids I, 2015). The main purpose of this paper is to determine the simple algebras in $\mathbf {I}$. It is shown that there are exactly five (nontrivial) simple algebras in $\mathbf {I}$. From this description we deduce that the semisimple subvarieties of $\mathbf {I}$ are precisely the subvarieties of the variety generated by these simple I-zroupoids and that they are locally finite. It also follows that the lattice of semisimple subvarieties of $\mathbf {I}$ is isomorphic to the direct product of a 4-element Boolean lattice and a 4-element chain.

Appendix

Proof of Lemma 3.3 As mentioned before, Items from (1) to (19) are proved in Cornejo and Sankappanavar (2015). Let $a,b,c,d \in A$.

• (20)
  $$\begin{aligned} \begin{array}{llll} &{}&{} [(0 \rightarrow a) \rightarrow b] \rightarrow a&{} \\ &{}&{}\quad = [(b \rightarrow a) \rightarrow (0 \rightarrow a)']' &{}\quad \text{ by } \text{(14) } \\ &{}&{}\quad =\! (b \rightarrow a)'' &{}\quad \text{ by } \text{(15) } \text{ using } x \!=\! b, y \!=\! a \\ &{}&{}\quad = b \rightarrow a \end{array} \end{aligned}$$
• (21)
  $$\begin{aligned} \begin{array}{llll} &{}&{} a \rightarrow (0 \rightarrow b) = [\{0 \rightarrow (0 \rightarrow b)\} &{} \\ &{}&{}\quad \rightarrow a] \rightarrow (0 \rightarrow b) &{} \text{ by } \text{(20) } \text{ with } x = 0 \rightarrow b, y = a \\ &{}&{}\quad = [(0 \rightarrow b) \rightarrow a] \rightarrow (0 \rightarrow b) &{}\quad \text{ by } \text{(7) } \\ &{}&{}\quad = (0 \rightarrow a) \rightarrow (0 \rightarrow b) &{} \quad \text{ by } \text{(6) } \end{array} \end{aligned}$$
• (22)
  $$\begin{aligned} \begin{array}{lcll} &{}&{} (a \rightarrow b)' \rightarrow (0 \rightarrow c) &{}\\ &{}&{}\quad = [(a \rightarrow b) \rightarrow 0] \rightarrow (0 \rightarrow c) &{} \\ &{}&{}\quad = [\{(0 \rightarrow c)' \rightarrow (a \rightarrow b)\} \rightarrow \{0 \rightarrow (0 \rightarrow c)\}']' &{} \; \hbox {by (I)} \\ &{}&{}\quad = [\{(0 \rightarrow c)' \rightarrow (a \rightarrow b)\} \rightarrow (0 \rightarrow c)']' &{} \; \hbox {by (7)} \\ &{}&{}\quad = [\{0 \rightarrow (a \rightarrow b)\} \rightarrow (0 \rightarrow c)']' &{}\; \hbox {by (6) taking }\\ &{} &{} &{}\, x = a \rightarrow b,\\ &{} &{} &{}\, y = (0 \rightarrow c)' \\ &{}&{}\quad = [\{0 \rightarrow (a \rightarrow b)\} \rightarrow (c' \rightarrow 0')']' &{}\; \hbox {by Lemma 3.1 (a)} \\ &{}&{}\quad = [(a \rightarrow b) \rightarrow c'] \rightarrow 0' &{} \quad \hbox {from (I)} \\ &{}&{}\quad = 0 \rightarrow [(a \rightarrow b) \rightarrow c']' &{}\;\hbox {by Lemma 3.1 (a)} \end{array} \end{aligned}$$
• (23)
  $$\begin{aligned}&[(a' \rightarrow b') \rightarrow (b \rightarrow a)'] \rightarrow (b \rightarrow a)\\&\quad = [(a' \rightarrow b') \rightarrow (b \rightarrow a)']'' \rightarrow (b \rightarrow a) \\&\quad = [(b' \rightarrow b) \rightarrow a]' \rightarrow (b \rightarrow a) \quad \text{ by } \text{(I) } \\&\quad = [b \rightarrow a]' \rightarrow (b \rightarrow a) \quad \text{ by } \text{ Lemma } \text{3.2 } \text{(d) } \\&\quad = b \rightarrow a \quad \text{ by } \text{ Lemma } \text{3.2 } \text{(d) } \end{aligned}$$
• (24)
  $$\begin{aligned} \begin{array}{llll} &{}&{}[0 \rightarrow (a' \rightarrow b')'] \rightarrow (b \rightarrow a) &{}\\ &{}&{}\quad = [(a' \rightarrow b') \rightarrow 0'] \rightarrow (b \rightarrow a) &{} \quad \text{ by } \text{ Lemma } \text{3.1 } \text{(a) } \\ &{}&{}\quad = [(a' \rightarrow b') \rightarrow (b \rightarrow a)'] \rightarrow (b \rightarrow a) &{}\quad \text{ by } \text{(1) } \\ &{}&{}\quad = b \rightarrow a &{} \quad \text{ from } \text{(23) } \end{array} \end{aligned}$$
• (25)
  $$\begin{aligned} \begin{array}{llll} &{}&{}(0 \rightarrow (a \rightarrow b)) \rightarrow a&{}\\ &{}&{}\quad = [(a' \rightarrow 0) \rightarrow \{(a \rightarrow b) \rightarrow a\}']' &{} \quad \text{ from } \text{(I) } \\ &{}&{}\quad = [(a' \rightarrow 0) \rightarrow \{(0 \rightarrow b) \rightarrow a\}']' &{} \quad \text{ by } \text{(6) } \\ &{}&{}\quad = [\{0 \rightarrow (0 \rightarrow b\}] \rightarrow a &{}\quad \text{ using } \text{(I) } \\ &{}&{}\quad = (0 \rightarrow b) \rightarrow a &{} \quad \text{ by } \text{(7) } \end{array} \end{aligned}$$
• (26)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}(0 \rightarrow a) \rightarrow (a \rightarrow b)&{}\\ &{}&{}\quad = [(a \rightarrow b) \rightarrow a] \rightarrow (a \rightarrow b) &{} \; \hbox {by (6) with } \\ &{} &{} &{}\quad y = a \rightarrow b, x = a \\ &{}&{}\quad = [a \rightarrow (b \rightarrow a)']' \rightarrow (a \rightarrow b) &{} \; \hbox {by (5)} \\ &{}&{}\quad = [a'' \rightarrow (b \rightarrow a)']' \rightarrow (a \rightarrow b) &{} \\ &{}&{}\quad = [(a' \rightarrow 0) \rightarrow (b \rightarrow a)']' \rightarrow (a \rightarrow b) &{} \\ &{}&{}\quad = [(0 \rightarrow b) \rightarrow a] \rightarrow (a \rightarrow b) &{}\; \hbox {from (I)} \\ &{}&{}\quad = [\{0 \rightarrow (a \rightarrow b)\} \rightarrow a] \rightarrow (a \rightarrow b) &{}\; \hbox {by (25)} \\ &{}&{}\quad = a \rightarrow (a \rightarrow b) &{}\;\hbox {by (20) with }\,\, x = a \rightarrow b, \\ &{} &{} &{} \quad \text {and }\,\, y = a \end{array} \end{aligned}$$
• (27)
  $$\begin{aligned} \begin{array}{llll} &{}&{}[b \rightarrow (0 \rightarrow a)]' &{}\\ &{}&{}\quad = [(0 \rightarrow b) \rightarrow (0 \rightarrow a)]' &{}\quad \text{ by } \text{(21) } \\ &{}&{}\quad = [\{(0 \rightarrow a)' \rightarrow 0\} \rightarrow \{b \rightarrow (0 \rightarrow a)\}']'' &{}\quad \text{ from } \text{(I) } \\ &{}&{}\quad = [(0 \rightarrow a)' \rightarrow 0] \rightarrow [b \rightarrow (0 \rightarrow a)]' &{} \\ &{}&{}\quad = (0 \rightarrow a)'' \rightarrow [b \rightarrow (0 \rightarrow a)]' &{} \\ &{}&{}\quad = (0 \rightarrow a) \rightarrow [b \rightarrow (0 \rightarrow a)]' &{} \end{array} \end{aligned}$$
• (28)
  $$\begin{aligned} \begin{array}{llll} &{}&{}[(a \rightarrow b) \rightarrow (0 \rightarrow a)]'&{}\\ &{}&{}\quad = [\{(0 \rightarrow a)' \rightarrow a\} \rightarrow \{b \rightarrow (0 \rightarrow a)\}']'' &{}\; \hbox {from (I)} \\ &{}&{}\quad = [(0 \rightarrow a)' \rightarrow a] \rightarrow [b \rightarrow (0 \rightarrow a)]' &{} \\ &{}&{}\quad = [\{(0 \rightarrow a) \rightarrow 0\} \rightarrow a] \rightarrow [b \rightarrow (0 \rightarrow a)]' &{} \\ &{}&{}\quad = (0 \rightarrow a) \rightarrow [b \rightarrow (0 \rightarrow a)]' &{} \; \hbox {by (20) with }\,\, x = a, \\ &{} &{} &{} \quad \text {and } y = 0 \end{array} \end{aligned}$$
• (29)
  $$\begin{aligned} \begin{array}{llll} &{}&{}b \rightarrow (0 \rightarrow a) &{}\\ &{}&{}\quad = [b \rightarrow (0 \rightarrow a)]'' &{} \\ &{}&{}\quad = [(0 \rightarrow a) \rightarrow \{b \rightarrow (0 \rightarrow a)\}']' &{} \quad \text{ by } \text{(27) } \\ &{}&{}\quad = [(a \rightarrow b) \rightarrow (0 \rightarrow a)]'' &{}\quad \text{ by } \text{(28) } \\ &{}&{}\quad = (a \rightarrow b) \rightarrow (0 \rightarrow a). &{} \end{array} \end{aligned}$$
• (30)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}b \rightarrow (b \rightarrow a)&{}\\ &{}&{}\quad = (0 \rightarrow b) \rightarrow (b \rightarrow a) &{} \quad \text{ by } \text{(26) } \\ &{}&{}\quad = [0 \rightarrow (0 \rightarrow b)] \rightarrow (b \rightarrow a) &{} \quad \text{ by } \text{(7) } \\ &{}&{}\quad = [(b \rightarrow a) \rightarrow (0 \rightarrow b)] \rightarrow (b \rightarrow a) &{} \quad \text{ by } \text{(6) } \\ &{}&{}\quad = [a \rightarrow (0 \rightarrow b)] \rightarrow (b \rightarrow a) &{} \quad \text{ by } \text{(29) } \end{array} \end{aligned}$$
• (31)
  $$\begin{aligned} \begin{array}{llll} a \rightarrow b &{} = &{} [0 \rightarrow (b' \rightarrow a')'] \rightarrow (a \rightarrow b) &{} \quad \text{ by } \text{(24) } \\ &{} = &{} [0 \rightarrow \{(b \rightarrow 0) \rightarrow a'\}'] \rightarrow (a \rightarrow b) &{} \\ &{} = &{} [(b \rightarrow 0)' \rightarrow (0 \rightarrow a)] \rightarrow (a \rightarrow b) &{}\quad \text{ by } \text{(22) } \\ &{} = &{} [b'' \rightarrow (0 \rightarrow a)] \rightarrow (a \rightarrow b) &{} \\ &{} = &{} [b \rightarrow (0 \rightarrow a)] \rightarrow (a \rightarrow b) &{} \\ &{} = &{} a \rightarrow (a \rightarrow b) &{} \quad \text{ by } \text{(30) } \end{array} \end{aligned}$$
• (32)
  $$\begin{aligned} \begin{array}{llll} &{}&{}c \rightarrow ((a \rightarrow b) \rightarrow c)'&{}\\ &{}&{}\quad = [\{c \rightarrow (a \rightarrow b)\} \rightarrow c]' &{} \quad \text{ by } \text{(5) } \\ &{}&{}\quad = [\{0 \rightarrow (a \rightarrow b)\} \rightarrow c]' &{}\quad \text{ by } \text{(6) } \\ &{}&{}\quad = [\{a \rightarrow (0 \rightarrow b)\} \rightarrow c]' &{}\quad \text{ by } \text{(13) } \end{array} \end{aligned}$$
• (33)
  $$\begin{aligned} \begin{array}{llll} &{}&{}[0 \rightarrow (a \rightarrow b)'] \rightarrow b&{}\\ &{}&{}\quad = [(a \rightarrow b) \rightarrow 0'] \rightarrow b &{} \quad \text{ by } \text{ Lemma } \text{3.1 } \text{(a) } \\ &{}&{}\quad = [(a \rightarrow b) \rightarrow b'] \rightarrow b &{} \quad \text{ by } \text{(1) } \\ &{}&{}\quad = [(a \rightarrow 0') \rightarrow b'] \rightarrow b &{}\quad \text{ by } \text{(1) } \\ &{}&{}\quad = [(a \rightarrow 0') \rightarrow 0'] \rightarrow b &{} \quad \text{ by } \text{(1) } \\ &{}&{}\quad = [(0 \rightarrow a') \rightarrow 0'] \rightarrow b &{} \quad \text{ by } \text{ Lemma } \text{3.1 } \text{(a) } \\ &{}&{}\quad = [0 \rightarrow (0 \rightarrow a')'] \rightarrow b &{}\quad \text{ by } \text{ Lemma } \text{3.1 } \text{(a) } \\ &{}&{}\quad = (0 \rightarrow a'') \rightarrow b &{} \quad \text{ by } \text{(11) } \\ &{}&{}\quad = (0 \rightarrow a) \rightarrow b &{} \end{array} \end{aligned}$$
• (34)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}[\{a \rightarrow (0 \rightarrow (b \rightarrow c)')\} \rightarrow c]' &{}\\ &{}&{}\quad = (c' \rightarrow a) \rightarrow [\{0 \rightarrow (b \rightarrow c)'\} \rightarrow c]' &{}\quad \text{ by } \text{(I) } \\ &{}&{}\quad = (c' \rightarrow a) \rightarrow [(0 \rightarrow b) \rightarrow c]' &{} \quad \text{ by } \text{(33) } \\ &{}&{}\quad = (c' \rightarrow a) \rightarrow [(c \rightarrow b) \rightarrow c]' &{} \quad \text{ by } \text{(6) } \\ &{}&{}\quad = (c' \rightarrow a) \rightarrow [c \rightarrow (b \rightarrow c)']'' &{}\quad \text{ by } \text{(5) } \\ &{}&{}\quad = (c' \rightarrow a) \rightarrow [c \rightarrow (b \rightarrow c)'] &{} \end{array} \end{aligned}$$
• (35)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}a \rightarrow [\{b \rightarrow (c \rightarrow a)'\} \rightarrow a]' &{}\\ &{}&{}\quad = [\{b \rightarrow (0 \rightarrow (c \rightarrow a)')\} \rightarrow a]' &{} \quad \text{ by } \text{(32) } \text{ with } x = b, \\ &{} &{} &{} y = (c \rightarrow a)', z = a \\ &{}&{}\quad = (a' \rightarrow b) \rightarrow \{a \rightarrow (c \rightarrow a)'\} &{} \quad \text{ by } \text{(34) } \text{ with } x = b,\\ &{} &{} &{}\quad y = c, z = a \end{array} \end{aligned}$$
• (36)
  $$\begin{aligned} \begin{array}{llll} &{}&{}\{0 \rightarrow (a \rightarrow b)\} \rightarrow b' &{}\\ &{}&{}\quad = [(b \rightarrow 0) \rightarrow \{(a \rightarrow b) \rightarrow b'\}']' &{} \quad \text{ from } \text{(I) } \\ &{}&{}\quad = [(b \rightarrow 0) \rightarrow \{(a \rightarrow 0') \rightarrow b'\}']' &{} \quad \text{ by } \text{(1) } \\ &{}&{}\quad = \{0 \rightarrow (a \rightarrow 0')\} \rightarrow b' &{} \quad \text{ from } \text{(I) } \\ &{}&{}\quad = \{0 \rightarrow (0 \rightarrow a')\} \rightarrow b' &{}\quad \text{ by } \text{ Lemma } \text{3.1 } \text{(a) } \\ &{}&{}\quad = (0 \rightarrow a') \rightarrow b' &{} \quad \text{ by } \text{(7) } \\ &{}&{}\quad = (a \rightarrow 0') \rightarrow b' &{} \quad \text{ by } \text{ Lemma } \text{3.1 } \text{(a) } \\ &{}&{}\quad = (a \rightarrow b'') \rightarrow b' &{} \quad \text{ by } \text{(1) } \\ &{}&{}\quad = (a \rightarrow b) \rightarrow b' &{} \\ &{}&{}\quad = b \rightarrow (a \rightarrow b)' &{} \quad \text{ by } \text{(17) } \end{array} \end{aligned}$$
• (37)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}(a' \rightarrow b) \rightarrow \{a \rightarrow (c \rightarrow a)'\}&{}\\ &{}&{}\quad = a \rightarrow [\{b \rightarrow (c \rightarrow a)'\} \rightarrow a]' &{} \quad \text{ by } \text{(35) } \\ &{}&{}\quad = (0 \rightarrow b) \rightarrow [\{0 \rightarrow (c \rightarrow a)\} \rightarrow a'] &{} \quad \text{ by } \text{(16) } \text{ with } x = a,\\ &{} &{} &{}\quad y = b, z = c \rightarrow a \\ &{}&{}\quad = (0 \rightarrow b) \rightarrow [a \rightarrow (c \rightarrow a)'] &{} \quad \text{ by } \text{(36) } \text{ with } \,\, x = c, \\ &{} &{} &{}\quad \text {and } y = a \end{array} \end{aligned}$$
• (38)
  $$\begin{aligned} \begin{array}{llll} &{}&{}a \rightarrow [(c \rightarrow b) \rightarrow a]'&{}\\ &{}&{}\quad = [\{c \rightarrow (0 \rightarrow b)\} \rightarrow a]' &{} \;\, \hbox {by (32) with }x = c, y = b,\\ &{} &{} &{}\quad \text {and }\,\, z = a \\ &{}&{}\quad = [\{(b \rightarrow c) \rightarrow (0 \rightarrow b)\} \rightarrow a]' &{} \;\, \hbox {by (29)} \\ &{}&{}\quad = (a' \rightarrow (b \rightarrow c)) \rightarrow [(0 \rightarrow b) \rightarrow a]' &{} \;\, \hbox {by (I)} \\ &{}&{}\quad = \{a' \rightarrow (b \rightarrow c)\} \rightarrow [(a \rightarrow b) \rightarrow a]' &{} \;\, \hbox {by (6)} \\ &{}&{}\quad = \{a' \rightarrow (b \rightarrow c)\} \rightarrow [a \rightarrow (b \rightarrow a)']'' &{} \;\, \hbox {by (5)} \\ &{}&{}\quad = \{a' \rightarrow (b \rightarrow c)\} \rightarrow [a \rightarrow (b \rightarrow a)'] &{} \end{array} \end{aligned}$$
• (39)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}a \rightarrow [(b \rightarrow c) \rightarrow a]'&{}\\ &{}&{}\quad = [a' \rightarrow (c \rightarrow b)] \rightarrow [a \rightarrow (c \rightarrow a)'] &{} \quad \text{ by } \text{(38) } \text{ with } x = a, y = c, \\ &{} &{} &{} \quad \text {and } z = b \\ &{}&{}\quad = [0 \rightarrow (c \rightarrow b)] \rightarrow [a \rightarrow (c \rightarrow a)'] &{}\quad \text{ by } \text{(37) } \text{ with } \,\, x = a,\\ &{} &{} &{}\quad y = c \rightarrow b, z = c \end{array} \end{aligned}$$
• (40)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}[0 \rightarrow (a \rightarrow b)] \rightarrow [c \rightarrow (a \rightarrow c)']&{}\\ &{}&{}\quad = c \rightarrow [(b \rightarrow a) \rightarrow c]' &{}\quad \text{ by } \text{(39) } \text{ with } x = c,\\ &{} &{} &{}\quad y = b, z = a \\ &{}&{}\quad = [\{c \rightarrow (b \rightarrow a)\} \rightarrow c]' &{}\quad \text{ by } \text{(5) } \\ &{}&{}\quad = [\{0 \rightarrow (b \rightarrow a)\} \rightarrow c]' &{} \quad \text{ by } \text{(6) } \\ &{}&{}\quad = [\{b \rightarrow (0 \rightarrow a)\} \rightarrow c]' &{} \quad \text{ by } \text{(13) } \\ &{}&{}\quad = (c' \rightarrow b) \rightarrow [(0 \rightarrow a) \rightarrow c]' &{}\quad \text{ from } \text{(I) } \\ &{}&{}\quad = (c' \rightarrow b) \rightarrow [(c \rightarrow a) \rightarrow c]' &{}\quad \text{ by } \text{(6) } \\ &{}&{}\quad = (c' \rightarrow b) \rightarrow [c \rightarrow (a \rightarrow c)'] &{}\quad \text{ by } \text{(5) } \\ &{}&{}\quad = (0 \rightarrow b) \rightarrow [c \rightarrow (a \rightarrow c)'] &{} \quad \text{ by } \text{(37) } \text{ with } \,\, x = c, \\ &{} &{} &{}\quad y = b, z = a \end{array} \end{aligned}$$
• (41)
  $$\begin{aligned}&a \rightarrow [(b \rightarrow c) \rightarrow a]'\\&\quad = [0 \rightarrow (c \rightarrow b)] \rightarrow [a \rightarrow (c \rightarrow a)'] \quad \text{ by } \text{(39) } \\&\quad = (0 \rightarrow b) \rightarrow [a \rightarrow (c \rightarrow a)']\\&\quad \text{ by } \text{(40) } \text{ with } \,\, x = c,\,\, y = b,\,\, z = a\\ \end{aligned}$$

• (42) This follows immediately from (18).

• (43)
  $$\begin{aligned} \begin{array}{lcll} a \rightarrow (b \rightarrow a)' &{} = &{} [(a \rightarrow b) \rightarrow a]' &{}\quad \text{ by } \text{(5) } \\ &{} = &{} [(0 \rightarrow b) \rightarrow a]' &{} \quad \text{ by } \text{(6) } \\ &{} = &{} [(b' \rightarrow 0') \rightarrow a]' &{} \quad \text{ by } \text{ Lemma } \text{3.1 } \text{(a) } \\ &{} = &{} (0 \rightarrow b') \rightarrow a' &{} \quad \text{ by } \text{(19) } \\ &{} = &{} (b \rightarrow 0') \rightarrow a' &{} \quad \text{ by } \text{ Lemma } \text{3.1 } \text{(a) } \end{array} \end{aligned}$$
• (44)
  $$\begin{aligned} \begin{array}{lcll} (a' \rightarrow b) \rightarrow a' &{} = &{} [a' \rightarrow (b \rightarrow a')']' &{}\quad \text{ by } \text{(5) } \\ &{} = &{} [(b \rightarrow 0') \rightarrow a'']' &{} \quad \text{ by } \text{(43) } \\ &{} = &{} [(b \rightarrow 0') \rightarrow a]' &{} \\ &{} = &{} [(0 \rightarrow b') \rightarrow a]' &{} \quad \text{ by } \text{ Lemma } \text{3.1 } \text{(b) } \\ &{} = &{} [(a \rightarrow b') \rightarrow a]' &{} \quad \text{ by } \text{(6) } \\ &{} = &{} a' &{} \quad \text{ by } \text{ Hypothesis } \end{array} \end{aligned}$$
• (45)
  $$\begin{aligned}&[\{b \rightarrow (0 \rightarrow c)\} \rightarrow (a \rightarrow 0')']'\\&\quad = [\{(a \rightarrow 0')'' \rightarrow b\} \rightarrow \{(0 \rightarrow c) \rightarrow (a \rightarrow 0')'\}']'' \quad \text{ by } \text{(I) } \\&\quad = [(a \rightarrow 0') \rightarrow b] \rightarrow [(0 \rightarrow c) \rightarrow (a \rightarrow 0')']' \quad \\&\quad = [(a \rightarrow 0') \rightarrow b] \rightarrow [(c \rightarrow a) \rightarrow 0'] \qquad \text{ by } \text{(I) } \\&\quad = [(a \rightarrow 0') \rightarrow b] \rightarrow [0 \rightarrow (c \rightarrow a)'] \qquad \text{ by } \text{ Lemma } \text{3.1 } \text{(b) } \\&\quad = [(0 \rightarrow a') \rightarrow b] \rightarrow [0 \rightarrow (c \rightarrow a)'] \qquad \text{ by } \text{ Lemma } \text{3.1 } \text{(b) } \end{aligned}$$
• (46)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}[\{a \rightarrow (0 \rightarrow b)\} \rightarrow c]'&{}\\ &{}&{}\quad = [(c' \rightarrow a) \rightarrow \{(0 \rightarrow b) \rightarrow c\}']'' &{}\quad \text{ by } \text{(I) } \\ &{}&{}\quad = (c' \rightarrow a) \rightarrow [(0 \rightarrow b) \rightarrow c]' &{} \\ &{}&{}\quad = (c' \rightarrow a) \rightarrow [(c \rightarrow b) \rightarrow c]' &{} \quad \text{ by } \text{(6) } \\ &{}&{}\quad = (c' \rightarrow a) \rightarrow [c \rightarrow (b \rightarrow c)']'' &{} \quad \text{ by } \text{(5) } \\ &{}&{}\quad = (c' \rightarrow a) \rightarrow [c \rightarrow (b \rightarrow c)'] &{} \\ &{}&{}\quad = (0 \rightarrow a) \rightarrow [c \rightarrow (b \rightarrow c)'] &{} \quad \text{ by } \text{(37) } \end{array} \end{aligned}$$
• (47)
  $$\begin{aligned}&[(0 \rightarrow a') \rightarrow b] \rightarrow [0 \rightarrow (c \rightarrow a)'] \\&\quad = [\{b \rightarrow (0 \rightarrow c)\} \rightarrow (a \rightarrow 0')']' \quad \text{ by } \text{(45) } \\&\quad = (0 \rightarrow b) \rightarrow [(a \rightarrow 0')' \rightarrow \{c \rightarrow (a \rightarrow 0')'\}'] \\&\quad \quad \quad \text{ by } \text{(46) } \text{ with } x = b, y = c, z = (a \rightarrow 0')' \\&\quad = (0 \rightarrow b) \rightarrow [(a \rightarrow 0')' \rightarrow \{c \rightarrow (a \rightarrow 0')'\}']'' \quad \\&\quad = (0 \rightarrow b) \rightarrow [\{(a \rightarrow 0')' \rightarrow c\} \rightarrow (a \rightarrow 0')']' \\&\quad \quad \quad \text{ by } \text{(5) } \\&\quad = (0 \rightarrow b) \rightarrow [(0 \rightarrow c) \rightarrow (a \rightarrow 0')']' \quad \text{ by } \text{(6) } \\&\quad = (0 \rightarrow b) \rightarrow [(c \rightarrow a) \rightarrow 0'] \quad \text{ by } \text{(I) } \\&\quad = (0 \rightarrow b) \rightarrow [0 \rightarrow (c \rightarrow a)'] \quad \text{ by } \text{ Lemma } \text{3.1 } \text{(b) } \\&\quad = b \rightarrow [0 \rightarrow (c \rightarrow a)'] \quad \text{ by } \text{(21) } \end{aligned}$$
• (48)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}[(0 \rightarrow a) \rightarrow b] \rightarrow (0 \rightarrow c)&{}\\ &{}&{}\quad = [0 \rightarrow \{(0 \rightarrow a) \rightarrow b\}] \rightarrow (0 \rightarrow c) &{} \quad \text{ by } \text{(21) } \\ &{}&{}\quad = [(0 \rightarrow a) \rightarrow (0 \rightarrow b)] \rightarrow (0 \rightarrow c) &{} \quad \text{ by } \text{(13) } \\ &{}&{}\quad = [a \rightarrow (0 \rightarrow b)] \rightarrow (0 \rightarrow c) &{} \quad \text{ by } \text{(21) } \\ &{}&{}\quad = [0 \rightarrow (a \rightarrow b)] \rightarrow (0 \rightarrow c) &{} \quad \text{ by } \text{(13) } \\ &{}&{}\quad = (a \rightarrow b) \rightarrow (0 \rightarrow c) &{} \quad \text{ by } \text{(21) } \\ &{}&{}\quad = 0 \rightarrow ((a \rightarrow b) \rightarrow c) &{} \quad \text{ by } \text{(13) } \end{array} \end{aligned}$$
• (49)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}b \rightarrow [0 \rightarrow (c \rightarrow a)'] &{}\\ &{}&{}\quad = [(0 \rightarrow a') \rightarrow b] \rightarrow [0 \rightarrow (c \rightarrow a)'] &{} \quad \text{ by } \text{(47) } \\ &{}&{}\quad = 0 \rightarrow [(a' \rightarrow b) \rightarrow (c \rightarrow a)'] &{} \quad \text{ by } \text{(48) } \\ &{}&{}\quad = (a' \rightarrow b) \rightarrow [0 \rightarrow (c \rightarrow a)'] &{} \quad \text{ by } \text{(13) } \end{array} \end{aligned}$$
• (50)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}b \rightarrow [0 \rightarrow (c \rightarrow a)']&{}\\ &{}&{}\quad = (a' \rightarrow b) \rightarrow [0 \rightarrow (c \rightarrow a)'] &{}\quad \text{ by } \text{(49) } \\ &{}&{}\quad = 0 \rightarrow [(a' \rightarrow b) \rightarrow (c \rightarrow a)'] &{} \quad \text{ by } \text{(13) } \end{array} \end{aligned}$$
• (51)
  $$\begin{aligned}&(0 \rightarrow a) \rightarrow [\{0 \rightarrow (b \rightarrow c)\} \rightarrow d']\\&\quad = (0 \rightarrow a) \rightarrow [\{0 \rightarrow (b \rightarrow c)\} \rightarrow \{d \rightarrow (0 \rightarrow d)'\}] \\&\quad \quad \quad \text{ by } \text{(3) } \\&\quad = (0 \rightarrow a) \rightarrow [d \rightarrow [(b \rightarrow c)' \rightarrow d]]' \\&\quad \quad \quad \text{ by } \text{(41) } \text{ with } x = d, y = b \rightarrow c, z = 0 \\&\quad = [\{a \rightarrow (0 \rightarrow (b \rightarrow c)')\}\rightarrow d]' \\&\quad \quad \quad \text{ by } \text{(46) } \text{ with } x = a, y = (b \rightarrow c)', z = d \\&\quad = [[0 \rightarrow \{(c' \rightarrow a) \rightarrow (b \rightarrow c)'\}] \rightarrow d]' \\&\quad \quad \quad \text{ by } \text{(50) } \text{ with } x = c, y = a, z = b \\&\quad = [[0 \rightarrow \{(c' \rightarrow a) \rightarrow (b \rightarrow c)'\}''] \rightarrow d]' \quad \\&\quad = [[0 \rightarrow \{(a \rightarrow b) \rightarrow c\}'] \rightarrow d]' \quad \text{ by } \text{(I) } \\&\quad = [[d \rightarrow [(a \rightarrow b) \rightarrow c]'] \rightarrow d]' \quad \text{ by } \text{(6) } \\&\quad = [d \rightarrow \{((a \rightarrow b) \rightarrow c)' \rightarrow d\}']'' \quad \text{ by } \text{(5) } \\&\quad = d \rightarrow [\{(a \rightarrow b) \rightarrow c\}' \rightarrow d]' \quad \\&\quad = [0 \rightarrow \{(a \rightarrow b) \rightarrow c\}] \rightarrow [d \rightarrow (0 \rightarrow d)'] \\&\quad \quad \text{ by } \text{(41) } \text{ with } x = d, y = (a \rightarrow b) \rightarrow c, z = 0 \\&\quad = [0 \rightarrow \{(a \rightarrow b) \rightarrow c\}] \rightarrow d' \quad \text{ by } \text{(3) } \end{aligned}$$
• (52)
  $$\begin{aligned}&(0 \rightarrow a) \rightarrow [(0 \rightarrow b') \rightarrow (0 \rightarrow c)'] \\&\quad = (0 \rightarrow a) \rightarrow [\{0 \rightarrow (b \rightarrow 0)\} \rightarrow (0 \rightarrow c)'] \quad \\&\quad = [0 \rightarrow \{(a \rightarrow b) \rightarrow 0\}] \rightarrow (0 \rightarrow c)' \\&\quad \quad \text{ by } \text{(51) } \text{ with } x = a, y = b, z = 0, u = 0 \rightarrow c \\&\quad = [0 \rightarrow (a \rightarrow b)'] \rightarrow (0 \rightarrow c)' \quad \\&\quad = [(a \rightarrow b) \rightarrow (0 \rightarrow c)]' \quad \text{ by } \text{(10) } \\&\quad = [0 \rightarrow \{(a \rightarrow b) \rightarrow c\}]' \quad \text{ by } \text{(13) } \end{aligned}$$
• (53)
  $$\begin{aligned}&0 \rightarrow [(a \rightarrow b) \rightarrow (c \rightarrow a')']\\&\quad = 0 \rightarrow [(a'' \rightarrow b) \rightarrow (c \rightarrow a')']''&\\&\quad = 0 \rightarrow [(b \rightarrow c) \rightarrow a']' \quad \text{ by } \text{(I) } \\&\quad = 0 \rightarrow [0 \rightarrow \{(b \rightarrow c) \rightarrow a'\}'] \quad \text{ by } \text{(7) } \\&\quad = 0 \rightarrow [(0 \rightarrow b) \rightarrow \{(0 \rightarrow c') \rightarrow (0 \rightarrow a')'\}] \\&\quad \quad \quad \text{ by } \text{(52) } \text{ with } x = b, y = c, z = a' \\&\quad = 0 \rightarrow [(0 \rightarrow b) \rightarrow \{(0 \rightarrow c') \rightarrow (a \rightarrow 0')'\}] \\&\quad \quad \quad \text{ by } \text{ Lemma } \text{3.1 } \text{(b) } \\&\quad = 0 \rightarrow [(0 \rightarrow b) \rightarrow \{(0 \rightarrow c') \rightarrow (a \rightarrow 0')'\}''] \quad \\&\quad = 0 \rightarrow [(0 \rightarrow b) \rightarrow \{(c' \rightarrow a) \rightarrow 0'\}'] \quad \text{ by } \text{(I) } \\&\quad = 0 \rightarrow [(0 \rightarrow b) \rightarrow \{(c' \rightarrow a) \rightarrow 0'\}']'' \quad \\&\quad = 0 \rightarrow [\{b \rightarrow (c' \rightarrow a)\} \rightarrow 0']' \quad \text{ by } \text{(I) } \\&\quad = 0 \rightarrow [0 \rightarrow \{b \rightarrow (c' \rightarrow a)\}']' \quad \text{ by } \text{ Lemma } \text{3.1 } \text{(b) } \\&\quad = 0 \rightarrow [b \rightarrow (c' \rightarrow a)]'' \quad \text{ by } \text{(11) } \\&\quad = 0 \rightarrow [b \rightarrow (c' \rightarrow a)] \quad \end{aligned}$$
• (54)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}0 \rightarrow [(a \rightarrow b)' \rightarrow c]&{}\\ &{}&{}\quad = 0 \rightarrow [(a \rightarrow b) \rightarrow c']' &{} \quad \text {by (8)} \\ &{}&{}\quad = 0 \rightarrow [(c'' \rightarrow a) \rightarrow (b \rightarrow c')']'' &{} \quad \text {by (I)} \\ &{}&{}\quad = 0 \rightarrow [(c \rightarrow a) \rightarrow (b \rightarrow c')'] &{} \text {} \\ &{}&{}\quad = 0 \rightarrow [a \rightarrow (b' \rightarrow c)] &{} \quad \text {by (53)} \end{array} \end{aligned}$$
• (55)
  $$\begin{aligned}&[\{(a \rightarrow b) \rightarrow c\} \rightarrow \{0 \rightarrow (b \rightarrow c)\}'] \rightarrow \{(a \rightarrow b) \rightarrow c\}\\&\quad = [0 \rightarrow \{0 \rightarrow (b \rightarrow c)\}'] \rightarrow \{(a \rightarrow b) \rightarrow c\} \quad \text {by (6)} \\&\quad = [0 \rightarrow (b \rightarrow c)'] \rightarrow [(a \rightarrow b) \rightarrow c] \quad \text {by (7)} \\&\quad = [0 \rightarrow (b \rightarrow c)'] \rightarrow [(c' \rightarrow a) \rightarrow (b \rightarrow c)']' \quad \text {by (I)} \\&\quad = [(c' \rightarrow a) \rightarrow (b \rightarrow c)']' \quad \text {by (4)} \\&\quad = (a \rightarrow b) \rightarrow c \quad \text {by (I)} \end{aligned}$$
• (56)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}[0 \rightarrow (a \rightarrow b)'] \rightarrow (c \rightarrow b) &{}\\ &{}&{}\quad = [0 \rightarrow (a'' \rightarrow b)'] \rightarrow (c \rightarrow b) &{} \\ &{}&{}\quad = [a' \rightarrow (0 \rightarrow b')] \rightarrow (c \rightarrow b) &{} \quad \text{ by } \text{(12) } \\ &{}&{}\quad = [0 \rightarrow \{(0 \rightarrow a') \rightarrow b'\}] \rightarrow (c \rightarrow b) &{} \quad \text{ by } \text{(42) } \\ &{}&{}\quad = [(0 \rightarrow a') \rightarrow (0 \rightarrow b')] \rightarrow (c \rightarrow b) &{} \quad \text{ by } \text{(13) } \\ &{}&{}\quad = [\{(c \rightarrow b)' \rightarrow (0 \rightarrow a')\} \\ &{}&{} \qquad \rightarrow \{(0 \rightarrow b') \rightarrow (c \rightarrow b)\}']' &{} \quad \text{ by } \text{(I) } \\ &{}&{}\quad = [\{(c \rightarrow b)' \rightarrow (0 \rightarrow a')\} \rightarrow (c \rightarrow b)']' &{} \quad \text{ by } \text{(2) } \\ &{}&{}\quad = [(c \rightarrow b)' \rightarrow \{(0 \rightarrow a') \rightarrow (c \rightarrow b)'\}']'' &{}\quad \text{ by } \text{(5) } \\ &{}&{}\quad = (c \rightarrow b)' \rightarrow [(0 \rightarrow a') \rightarrow (c \rightarrow b)']' &{} \\ &{}&{}\quad = [(0 \rightarrow a') \rightarrow 0'] \rightarrow (c \rightarrow b)'' &{} \quad \text{ by } \text{(43) } \\ &{}&{}\quad = [(0 \rightarrow a') \rightarrow 0'] \rightarrow (c \rightarrow b) &{} \\ &{}&{}\quad = [0 \rightarrow (0 \rightarrow a')'] \rightarrow (c \rightarrow b) &{} \quad \text{ by } \text{ Lemma } \text{3.1 } \text{(b) } \\ &{}&{}\quad = (0 \rightarrow a'') \rightarrow (c \rightarrow b) &{}\quad \text{ by } \text{(11) } \\ &{}&{}\quad = (0 \rightarrow a) \rightarrow (c \rightarrow b) &{} \end{array} \end{aligned}$$
• (57)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}[b \rightarrow [0 \rightarrow (0 \rightarrow c)']] \rightarrow a&{}\\ &{}&{}\quad = [b \rightarrow (0 \rightarrow c')] \rightarrow a &{} \quad \text{ by } \text{(11) } \\ &{}&{}\quad = [0 \rightarrow (b' \rightarrow c)'] \rightarrow a &{} \quad \text{ by } \text{(12) } \\ &{}&{}\quad = [a \rightarrow (b' \rightarrow c)'] \rightarrow a &{} \quad \text{ by } \text{(6) } \end{array} \end{aligned}$$
• (58)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}b \rightarrow (c \rightarrow a) &{}\\ &{}&{}\quad = [\{0 \rightarrow (c \rightarrow a)\} \rightarrow b] \rightarrow (c \rightarrow a) &{} \quad \text{ by } \text{(20) } \\ &{}&{}\quad = [[(c \rightarrow a)' \rightarrow [0 \rightarrow (c \rightarrow a)]] \rightarrow [b \rightarrow (c \rightarrow a)]']' &{} \quad \text{ by } \text{(I) } \\ &{}&{}\quad = [[(c \rightarrow a)' \rightarrow [(c \rightarrow a)' \rightarrow 0']] \rightarrow [b \rightarrow (c \rightarrow a)]']' &{} \quad \text{ by } \text{ Lemma } \text{3.1 } \text{(b) } \\ &{}&{}\quad = [[(c \rightarrow a)' \rightarrow 0'] \rightarrow [b \rightarrow (c \rightarrow a)]']' &{} \quad \text{ by } \text{(31) } \\ &{}&{}\quad = [[0 \rightarrow (c \rightarrow a)] \rightarrow [b \rightarrow (c \rightarrow a)]']' &{} \quad \text{ by } \text{ Lemma } \text{3.1 } \text{(b) } \\ &{}&{}\quad = [[c \rightarrow (0 \rightarrow a)] \rightarrow [b \rightarrow (c \rightarrow a)]']' &{} \quad \text{ by } \text{(13) } \\ &{}&{}\quad = [[c'' \rightarrow (0 \rightarrow a)]'' \rightarrow [b \rightarrow (c \rightarrow a)]']' &{} \\ &{}&{}\quad = [[(0 \rightarrow c') \rightarrow (0 \rightarrow a)']' \rightarrow [b \rightarrow (c \rightarrow a)]']' &{} \quad \text{ by } \text{(10) } \\ &{}&{}\quad = [[(0 \rightarrow (0 \rightarrow c')) \rightarrow (0 \rightarrow a)']' \rightarrow [b \rightarrow (c \rightarrow a)]']' &{} \quad \text{ by } \text{(31) } \\ &{}&{}\quad = [[((0 \rightarrow a)' \rightarrow (0 \rightarrow c')) \rightarrow (0 \rightarrow a)']' \rightarrow [b \rightarrow (c \rightarrow a)]']' \\ &{}&{} &{} \quad \text{ by } \text{(6) } \\ &{}&{}\quad = [[((0 \rightarrow a)' \rightarrow (0 \rightarrow c')) \rightarrow (0 \rightarrow (0 \rightarrow a))']' \\ &{}&{}\quad \rightarrow [b \rightarrow (c \rightarrow a)]']' &{}\quad \text{ by } \text{(31) } \\ &{}&{}\quad = [[(0 \rightarrow c')' \rightarrow (0 \rightarrow a)] \rightarrow [b \rightarrow (c \rightarrow a)]']' &{} \quad \text{ by } \text{(I) } \\ &{}&{}\quad = [[(0 \rightarrow c')' \rightarrow (0 \rightarrow a'')] \rightarrow [b \rightarrow (c \rightarrow a)]']' &{} \\ &{}&{}\quad = [[0 \rightarrow [(0 \rightarrow c')'' \rightarrow a']'] \rightarrow [b \rightarrow (c \rightarrow a)]']' &{} \quad \text{ by } \text{(12) } \\ &{}&{}\quad = [[0 \rightarrow [(0 \rightarrow c') \rightarrow a']'] \rightarrow [b \rightarrow (c \rightarrow a)]']' &{} \\ &{}&{}\quad = [[0 \rightarrow [(c \rightarrow 0') \rightarrow a']'] \rightarrow [b \rightarrow (c \rightarrow a)]']' &{} \quad \text{ by } \text{ Lemma } \text{3.1 } \text{(b) } \\ &{}&{}\quad = [[0 \rightarrow [(c \rightarrow a'') \rightarrow a']'] \rightarrow [b \rightarrow (c \rightarrow a)]']' &{} \quad \text{ by } \text{(1) } \\ &{}&{}\quad = [[0 \rightarrow [(c \rightarrow a) \rightarrow a']'] \rightarrow [b \rightarrow (c \rightarrow a)]']' &{} \\ &{}&{}\quad = [[(c \rightarrow a)' \rightarrow (0 \rightarrow a)] \rightarrow [b \rightarrow (c \rightarrow a)]']' &{} \quad \text{ by } \text{(22) } \\ &{}&{}\quad = [(0 \rightarrow a) \rightarrow b] \rightarrow (c \rightarrow a) &{} \quad \text{ by } \text{(I) } \end{array} \end{aligned}$$
• (59)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}a \rightarrow ((b \rightarrow a) \rightarrow b)&{}\\ &{}&{}\quad = a \rightarrow [(0 \rightarrow a) \rightarrow b] &{} \quad \text{ by } \text{(6) } \\ &{}&{}\quad = [(0 \rightarrow b) \rightarrow a] \rightarrow [(0 \rightarrow a) \rightarrow b] &{} \quad \text{ by } \text{(58) } \\ &{}&{}\quad = [(a \rightarrow b) \rightarrow a] \rightarrow [(b \rightarrow a) \rightarrow b] &{} \quad \text{ by } \text{(6) } \\ &{}&{}\quad = a \rightarrow b &{} \quad \text{ by } \text{(9) } \end{array} \end{aligned}$$
• (60)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}[(a \rightarrow b) \rightarrow (b \rightarrow c)]'&{}\\ &{}&{}\quad = [[(b \rightarrow c)' \rightarrow a] \rightarrow [b \rightarrow (b \rightarrow c)]']'' &{} \quad \text{ by } \text{(I) } \\ &{}&{}\quad = [(b \rightarrow c)' \rightarrow a] \rightarrow [b \rightarrow (b \rightarrow c)]' &{} \\ &{}&{}\quad = [(b \rightarrow c)' \rightarrow a] \rightarrow (b \rightarrow c)' &{} \quad \text{ by } \text{(31) } \\ &{}&{}\quad = [0 \rightarrow a] \rightarrow (b \rightarrow c)' &{} \quad \text{ by } \text{(6) } \end{array} \end{aligned}$$
• (61)
  $$\begin{aligned} \begin{array}{lcll} &{}&{}(a \rightarrow b) \rightarrow (b \rightarrow a) &{}\\ &{}&{}\quad = [[(b \rightarrow a)' \rightarrow a] \rightarrow [b \rightarrow (b \rightarrow a)]']' &{}\quad \text{ by } \text{(I) } \\ &{}&{}\quad = [[(b \rightarrow a)' \rightarrow a] \rightarrow [b \rightarrow a]']' &{} \quad \text{ by } \text{(31) } \\ &{}&{}\quad = [[0 \rightarrow a] \rightarrow [b \rightarrow a]']' &{}\quad \text{ by } \text{(6) } \\ &{}&{}\quad = (b \rightarrow a)'' &{} \quad \text{ by } \text{(4) } \\ &{}&{}\quad = b \rightarrow a &{} \end{array} \end{aligned}$$
• (62)
  $$\begin{aligned}&[[(a \rightarrow b) \rightarrow c] \rightarrow (c' \rightarrow a)'] \rightarrow [(a \rightarrow b) \rightarrow c] \\&\quad = [0 \rightarrow (c' \rightarrow a)'] \rightarrow [(a \rightarrow b) \rightarrow c] \quad \quad \text{ by } \text{(6) } \\&\quad = [(c' \rightarrow a) \rightarrow 0'] \rightarrow [(a \rightarrow b) \rightarrow c] \quad \quad \text{ by } \text{ Lemma } \text{3.1 } \text{(b) } \\&\quad = [(c' \rightarrow a) \rightarrow [(a \rightarrow b) \rightarrow c]'] \rightarrow [(a \rightarrow b) \rightarrow c] \quad \quad \text{ by } \text{(1) } \\&\quad = [(c' \rightarrow a) \rightarrow [(a \rightarrow b) \rightarrow c]']'' \rightarrow [(a \rightarrow b) \rightarrow c] \quad \quad \\&\quad = [(a \rightarrow (a \rightarrow b)) \rightarrow c]' \rightarrow [(a \rightarrow b) \rightarrow c] \quad \quad \text{ by } \text{(I) } \\&\quad = [(a \rightarrow b) \rightarrow c]' \rightarrow [(a \rightarrow b) \rightarrow c] \quad \quad \text{ by } \text{(31) } \\&\quad = (a \rightarrow b) \rightarrow c \quad \quad \text{ by } \text{ Lemma } \text{3.2 } \text{(d) } \end{aligned}$$
• (63)
  $$\begin{aligned}&[[(a \rightarrow b) \rightarrow c] \rightarrow [c' \rightarrow (b \rightarrow a)]'] \rightarrow [(a \rightarrow b) \rightarrow c] \\&\quad = \left[ [((b \rightarrow a) \rightarrow (a \rightarrow b)) \rightarrow c] \rightarrow [c' \rightarrow (b \rightarrow a)]'] \right. \\&\qquad \left. \rightarrow [((b \rightarrow a) \rightarrow (a \rightarrow b)) \rightarrow c\right] \quad \text {by (61)} \\&\quad = ((b \rightarrow a) \rightarrow (a \rightarrow b)) \rightarrow c \quad \text{ by } \text{(62) } \\&\quad = (a \rightarrow b) \rightarrow c \quad \text{ by } \text{(61) }. \end{aligned}$$

  $\square$