let B be B_Lattice; for a, b, c being Element of B st a <=> b = a <=> c holds
b = c
let a, b, c be Element of B; ( a <=> b = a <=> c implies b = c )
set ab = a "/\" b;
set ac = a "/\" c;
set bc = b "/\" c;
set b9c9 = (b `) "/\" (c `);
set a9b9 = (a `) "/\" (b `);
set a9c9 = (a `) "/\" (c `);
set a9b = (a `) "/\" b;
set a9c = (a `) "/\" c;
set ab9 = a "/\" (b `);
set ac9 = a "/\" (c `);
A1:
(a <=> b) <=> (a <=> c) = ((a <=> b) "/\" (a <=> c)) "\/" (((a <=> b) `) "/\" ((a <=> c) `))
by Th50;
A2:
( a <=> b = (a "/\" b) "\/" ((a `) "/\" (b `)) & a <=> c = (a "/\" c) "\/" ((a `) "/\" (c `)) )
by Th50;
A3:
(a <=> b) ` = (a "/\" (b `)) "\/" ((a `) "/\" b)
by Th51;
A4:
(a <=> c) ` = (a "/\" (c `)) "\/" ((a `) "/\" c)
by Th51;
A5:
((a "/\" b) "\/" ((a `) "/\" (b `))) "/\" ((a "/\" c) "\/" ((a `) "/\" (c `))) = ((a "/\" b) "/\" ((a "/\" c) "\/" ((a `) "/\" (c `)))) "\/" (((a `) "/\" (b `)) "/\" ((a "/\" c) "\/" ((a `) "/\" (c `))))
by LATTICES:def 11;
A6:
( (a "/\" b) "/\" ((a "/\" c) "\/" ((a `) "/\" (c `))) = ((a "/\" b) "/\" (a "/\" c)) "\/" ((a "/\" b) "/\" ((a `) "/\" (c `))) & (a "/\" b) "/\" ((a `) "/\" (c `)) = ((a "/\" b) "/\" (a `)) "/\" (c `) )
by LATTICES:def 7, LATTICES:def 11;
A7:
((a `) "/\" (b `)) "/\" ((a "/\" c) "\/" ((a `) "/\" (c `))) = (((a `) "/\" (b `)) "/\" (a "/\" c)) "\/" (((a `) "/\" (b `)) "/\" ((a `) "/\" (c `)))
by LATTICES:def 11;
A8:
(b "/\" a) "/\" (a `) = b "/\" (a "/\" (a `))
by LATTICES:def 7;
A9:
((a `) "/\" (b `)) "/\" (a "/\" c) = (((a `) "/\" (b `)) "/\" a) "/\" c
by LATTICES:def 7;
A10:
((b `) "/\" (a `)) "/\" a = (b `) "/\" ((a `) "/\" a)
by LATTICES:def 7;
A11:
((a "/\" (b `)) "\/" ((a `) "/\" b)) "/\" ((a "/\" (c `)) "\/" ((a `) "/\" c)) = ((a "/\" (b `)) "/\" ((a "/\" (c `)) "\/" ((a `) "/\" c))) "\/" (((a `) "/\" b) "/\" ((a "/\" (c `)) "\/" ((a `) "/\" c)))
by LATTICES:def 11;
A12:
( (a "/\" (b `)) "/\" ((a "/\" (c `)) "\/" ((a `) "/\" c)) = ((a "/\" (b `)) "/\" (a "/\" (c `))) "\/" ((a "/\" (b `)) "/\" ((a `) "/\" c)) & (a "/\" (b `)) "/\" ((a `) "/\" c) = ((a "/\" (b `)) "/\" (a `)) "/\" c )
by LATTICES:def 7, LATTICES:def 11;
A13:
((a `) "/\" b) "/\" ((a "/\" (c `)) "\/" ((a `) "/\" c)) = (((a `) "/\" b) "/\" (a "/\" (c `))) "\/" (((a `) "/\" b) "/\" ((a `) "/\" c))
by LATTICES:def 11;
A14:
((b `) "/\" a) "/\" (a `) = (b `) "/\" (a "/\" (a `))
by LATTICES:def 7;
A15:
( b "/\" (Bottom B) = Bottom B & (Bottom B) "/\" (c `) = Bottom B & (Bottom B) "/\" c = Bottom B & a "/\" (a `) = Bottom B & (a `) "/\" a = Bottom B & a "/\" (b `) = (b `) "/\" a & (a `) "/\" b = b "/\" (a `) & ((a `) "/\" b) "/\" (a "/\" (c `)) = (((a `) "/\" b) "/\" a) "/\" (c `) & (b "/\" (a `)) "/\" a = b "/\" ((a `) "/\" a) & ((a "/\" (b `)) "/\" (a "/\" (c `))) "\/" (Bottom B) = (a "/\" (b `)) "/\" (a "/\" (c `)) & (Bottom B) "\/" (((a `) "/\" b) "/\" ((a `) "/\" c)) = ((a `) "/\" b) "/\" ((a `) "/\" c) )
by LATTICES:20, LATTICES:def 7;
( (a "/\" b) "/\" (a "/\" c) = ((a "/\" b) "/\" a) "/\" c & (a "/\" b) "/\" a = a "/\" (a "/\" b) & a "/\" (a "/\" b) = (a "/\" a) "/\" b & a "/\" a = a & ((a `) "/\" (b `)) "/\" ((a `) "/\" (c `)) = (((a `) "/\" (b `)) "/\" (a `)) "/\" (c `) & ((a `) "/\" (b `)) "/\" (a `) = (a `) "/\" ((a `) "/\" (b `)) & (a `) "/\" ((a `) "/\" (b `)) = ((a `) "/\" (a `)) "/\" (b `) & (a `) "/\" (a `) = a ` & (a "/\" (b `)) "/\" (a "/\" (c `)) = ((a "/\" (b `)) "/\" a) "/\" (c `) & (a "/\" (b `)) "/\" a = a "/\" (a "/\" (b `)) & (a "/\" b) "/\" c = a "/\" (b "/\" c) & a "/\" (a "/\" (b `)) = (a "/\" a) "/\" (b `) & ((a `) "/\" b) "/\" ((a `) "/\" c) = (((a `) "/\" b) "/\" (a `)) "/\" c & ((a `) "/\" b) "/\" (a `) = (a `) "/\" ((a `) "/\" b) & ((a `) "/\" b) "/\" c = (a `) "/\" (b "/\" c) & (a "/\" (b `)) "/\" (c `) = a "/\" ((b `) "/\" (c `)) & ((a `) "/\" (b `)) "/\" (c `) = (a `) "/\" ((b `) "/\" (c `)) & (a `) "/\" ((a `) "/\" b) = ((a `) "/\" (a `)) "/\" b & ((a "/\" (b "/\" c)) "\/" ((a `) "/\" ((b `) "/\" (c `)))) "\/" ((a "/\" ((b `) "/\" (c `))) "\/" ((a `) "/\" (b "/\" c))) = (((a "/\" (b "/\" c)) "\/" ((a `) "/\" ((b `) "/\" (c `)))) "\/" (a "/\" ((b `) "/\" (c `)))) "\/" ((a `) "/\" (b "/\" c)) & ((a "/\" (b "/\" c)) "\/" ((a `) "/\" ((b `) "/\" (c `)))) "\/" (a "/\" ((b `) "/\" (c `))) = (a "/\" ((b `) "/\" (c `))) "\/" ((a "/\" (b "/\" c)) "\/" ((a `) "/\" ((b `) "/\" (c `)))) & (a "/\" ((b `) "/\" (c `))) "\/" ((a "/\" (b "/\" c)) "\/" ((a `) "/\" ((b `) "/\" (c `)))) = ((a "/\" ((b `) "/\" (c `))) "\/" (a "/\" (b "/\" c))) "\/" ((a `) "/\" ((b `) "/\" (c `))) & (a "/\" ((b `) "/\" (c `))) "\/" (a "/\" (b "/\" c)) = a "/\" (((b `) "/\" (c `)) "\/" (b "/\" c)) & ((b `) "/\" (c `)) "\/" (b "/\" c) = (b "/\" c) "\/" ((b `) "/\" (c `)) & ((a `) "/\" ((b `) "/\" (c `))) "\/" ((a `) "/\" (b "/\" c)) = (a `) "/\" (((b `) "/\" (c `)) "\/" (b "/\" c)) & (Top B) "/\" (((b `) "/\" (c `)) "\/" (b "/\" c)) = ((b `) "/\" (c `)) "\/" (b "/\" c) & ((a "/\" (((b `) "/\" (c `)) "\/" (b "/\" c))) "\/" ((a `) "/\" ((b `) "/\" (c `)))) "\/" ((a `) "/\" (b "/\" c)) = (a "/\" (((b `) "/\" (c `)) "\/" (b "/\" c))) "\/" (((a `) "/\" ((b `) "/\" (c `))) "\/" ((a `) "/\" (b "/\" c))) & a "\/" (a `) = Top B & (a "/\" (((b `) "/\" (c `)) "\/" (b "/\" c))) "\/" ((a `) "/\" (((b `) "/\" (c `)) "\/" (b "/\" c))) = (a "\/" (a `)) "/\" (((b `) "/\" (c `)) "\/" (b "/\" c)) )
by LATTICES:21, LATTICES:def 5, LATTICES:def 7, LATTICES:def 11;
then A16:
(a <=> b) <=> (a <=> c) = b <=> c
by A2, A1, Th50, A3, A4, A5, A6, A7, A8, A14, A15, A9, A10, A11, A12, A13;
assume A17:
a <=> b = a <=> c
; b = c
then A18:
(a <=> b) => (a <=> c) = Top B
by FILTER_0:28;
A19:
b <=> c = Top B
by A16, A17, A18;
then A20:
b "/\" (Top B) [= b "/\" (b => c)
by LATTICES:6, LATTICES:9;
A21:
c "/\" (Top B) [= c "/\" (c => b)
by A19, LATTICES:6, LATTICES:9;
A22:
b "/\" (b => c) [= c
by FILTER_0:def 7;
A23:
c "/\" (c => b) [= b
by FILTER_0:def 7;
A24:
b [= c
by A20, A22, LATTICES:7;
c [= b
by A21, A23, LATTICES:7;
hence
b = c
by A24, LATTICES:8; verum