reserve L,L1,L2 for Lattice,
  F1,F2 for Filter of L,
  p,q,r,s for Element of L,
  p1,q1,r1,s1 for Element of L1,
  p2,q2,r2,s2 for Element of L2,
  X,x,x1,x2,y,y1,y2 for set,
  D,D1,D2 for non empty set,
  R for Relation,
  RD for Equivalence_Relation of D,
  a,b,d for Element of D,
  a1,b1,c1 for Element of D1,
  a2,b2,c2 for Element of D2,
  B for B_Lattice,
  FB for Filter of B,
  I for I_Lattice,
  FI for Filter of I ,
  i,i1,i2,j,j1,j2,k for Element of I,
  f1,g1 for BinOp of D1,
  f2,g2 for BinOp of D2;
reserve F,G for BinOp of D,RD;
reserve B for B_Lattice,
  a,b,c,d for Element of B;

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;
