theorem Th52:
  a <=> b = a <=> c implies b = c
proof
  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 = ab"\/"a9b9 & a<=>c = ac"\/"a9c9
  by Th50;
A3: (a<=>b)` = ab9"\/"a9b
  by Th51;
A4:  (a<=>c)` = ac9"\/"a9c
  by Th51;
A5: (ab"\/"a9b9)"/\"(ac"\/"a9c9) = (ab"/\"(ac"\/"a9c9))"\/"(a9b9"/\"(ac"\/"
  a9c9))
  by LATTICES:def 11;
A6: ab"/\"(ac"\/"a9c9) = (ab"/\"ac)"\/"(ab"/\"a9c9) & ab"/\"a9c9 = ab"/\"a`"/\"
  c`
  by LATTICES:def 7,def 11;
A7: a9b9"/\"(ac"\/"a9c9) = (a9b9"/\"ac)"\/"(a9b9"/\"a9c9)
  by LATTICES:def 11;
A8:b"/\"a"/\"a` = b "/\"(a"/\"a`)
  by LATTICES:def 7;
A9:  a9b9"/\"ac = a9b9"/\"a"/\"c
  by LATTICES:def 7;
A10:  b`"/\"a`"/\"a = b`"/\"(a`"/\"a)
  by LATTICES:def 7;
A11: (ab9"\/"a9b)"/\"(ac9"\/"a9c) = (ab9"/\"(ac9"\/"a9c))"\/"(a9b"/\"(ac9"\/"
  a9c))
  by LATTICES:def 11;
A12
: ab9"/\"(ac9"\/"a9c) = (ab9"/\"ac9)"\/"(ab9"/\"a9c) & ab9"/\"a9c = ab9"/\"a`
  "/\"c
  by LATTICES:def 7,def 11;
A13:  a9b"/\"(ac9"\/"a9c) = (a9b"/\"ac9)"\/"(a9b"/\"a9c)
  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 & ab9 = b`"/\"a & a9b = b"/\"a` &
  a9b"/\"ac9 = a9b"/\"a"/\"c` & b"/\"a`"/\"a = b"/\"(a`"/\"a) &
  (ab9"/\"ac9)"\/"Bottom B = ab9"/\"ac9 & Bottom
  B"\/"(a9b"/\"a9c) = a9b"/\"a9c by LATTICES:20,def 7;
  ab"/\"ac = ab"/\"a"/\"c & ab"/\"a = a"/\"ab & a"/\"ab = a"/\"a"/\"b & a "/\"
  a = a &
  a9b9"/\"a9c9 = a9b9"/\"a`"/\"c` & a9b9"/\"a` = a`"/\"a9b9 & a`"/\"a9b9 = a`
  "/\"a`"/\"b` &
  a`"/\"a` = a` & ab9"/\"ac9 = ab9"/\"a"/\"c` & ab9"/\"a = a"/\"ab9 & (a"/\"b
  "/\"c) = a"/\"bc &
  a"/\"ab9 = a"/\"a"/\"b` & a9b"/\"a9c = a9b"/\"a`"/\"c & a9b"/\"a` = a`"/\"
  a9b &
  (a`"/\"b"/\"c) = a`"/\"bc & (a"/\"b`"/\"c`) = a"/\"b9c9 & (a`"/\"b`"/\"c`)
  = a`"/\"b9c9 & a`"/\"a9b = a`"/\"a`"/\"b &
  (a"/\"bc)"\/"(a`"/\"b9c9)"\/"((a"/\"b9c9)"\/"(a`"/\"bc)) =
  (a"/\"bc)"\/"(a`"/\"b9c9)"\/"(a"/\"b9c9)"\/"(a`"/\"bc) &
  (a"/\"bc)"\/"(a`"/\"b9c9)"\/"(a"/\"b9c9) = (a"/\"b9c9)"\/"((a"/\"bc)"\/"(a`
  "/\"b9c9)) &
  (a"/\"b9c9)"\/"((a"/\"bc)"\/"(a`"/\"b9c9)) = (a"/\"b9c9)"\/"(a"/\"bc)"\/"
  (a`"/\"b9c9) &
  (a"/\"b9c9)"\/"(a"/\"bc) = a"/\"(b9c9"\/"bc) & b9c9"\/"bc = bc"\/"b9c9 &
  (a`"/\"b9c9)"\/"(a`"/\"bc) = a`"/\"(b9c9"\/"bc) & (Top B)"/\"(b9c9"\/"
  bc) = b9c9"\/"bc & (a"/\"(b9c9"\/"bc))"\/"(a`"/\"b9c9)"\/"(a`"/\"bc) =
  (a"/\"(b9c9"\/"bc))"\/"((a`"/\"b9c9)"\/"(a`"/\"bc)) & a"\/"a` = Top B &
  (a"/\"(b9c9"\/"bc))"\/"(a`"/\"(b9c9"\/"bc)) = (a"\/"a`)"/\"(b9c9"\/"bc)
  by LATTICES:21,def 5,def 7,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;
  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:6,9;
A21:  c"/\"Top B [= c"/\"(c => b) by A19,LATTICES:6,9;
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 thesis by A24,LATTICES:8;
end;
