
theorem Th38: :: Theorem 2.11, p. 61-62, (2) implies (1)
:: CCL: Immediate from 2.8. (?)  One has to construct a homomorphism.
  for L being continuous complete LATTICE, R being Subset of [:L,
  L:] st R is Equivalence_Relation of the carrier of L & ex LR being continuous
complete LATTICE st the carrier of LR = Class EqRel R & for g being Function of
  L, LR st for x being Element of L holds g.x = Class(EqRel R, x) holds g is
  CLHomomorphism of L, LR holds subrelstr R is CLSubFrame of [:L, L:]
proof
  let L be continuous complete LATTICE, R be Subset of [:L, L:];
  assume R is Equivalence_Relation of the carrier of L;
  then
A1: EqRel R = R by Def1;
  set ER = EqRel R;
  given LR being continuous complete LATTICE such that
A2: the carrier of LR = Class EqRel R and
A3: for g being Function of L, LR st for x being Element of L holds g.x
  = Class(EqRel R, x) holds g is CLHomomorphism of L, LR;
  deffunc F(object) = Class(ER, $1);
  set CER = Class ER;
  set cL = the carrier of L, cLR = the carrier of LR;
A4: for x be object st x in cL holds F(x) in CER by EQREL_1:def 3;
  consider g being Function of cL, CER such that
A5: for x being object st x in cL holds g.x = F(x) from FUNCT_2:sch 2(A4);
  reconsider g as Function of L, LR by A2;
  set k = g;
A6: dom g = cL by FUNCT_2:def 1;
  now
    let x be object;
    hereby
      assume
A7:   x in R;
      then x in the carrier of [:L, L:];
      then x in [:cL, cL:] by YELLOW_3:def 2;
      then consider x1, x2 being object such that
A8:   x1 in cL & x2 in cL and
A9:   x = [x1, x2] by ZFMISC_1:def 2;
      reconsider x1, x2 as Element of L by A8;
A10:  k.x1 = Class(EqRel R, x1) & k.x2 = Class(EqRel R, x2) by A5;
      x1 in Class(EqRel R, x2) by A1,A7,A9,EQREL_1:19;
      then k.x1 = k.x2 by A10,EQREL_1:23;
      then
A11:  [k.x1, k.x2] in id cLR by RELAT_1:def 10;
      dom [:k, k:] = [:dom k, dom k:] by FUNCT_3:def 8;
      then
A12:  [x1, x2] in dom [:k, k:] by A6,ZFMISC_1:87;
      [:k, k:].(x1, x2) = [k.x1, k.x2] by A6,FUNCT_3:def 8;
      hence x in [:k, k:]"(id cLR) by A9,A11,A12,FUNCT_1:def 7;
    end;
    assume
A13: x in [:k, k:]"(id cLR);
    then
A14: [:k, k:].x in id cLR by FUNCT_1:def 7;
    x in dom [:k, k:] by A13,FUNCT_1:def 7;
    then x in [:dom k, dom k:] by FUNCT_3:def 8;
    then consider x1, x2 being object such that
A15: x1 in dom k & x2 in dom k and
A16: x = [x1, x2] by ZFMISC_1:def 2;
    reconsider x1, x2 as Element of L by A15;
A17: k.x1=Class(EqRel R, x1) & k.x2 = Class(EqRel R, x2) by A5;
    [:k, k:].(x1, x2) = [k.x1, k.x2] by A15,FUNCT_3:def 8;
    then k.x1 = k.x2 by A14,A16,RELAT_1:def 10;
    then x1 in Class(ER, x2) by A17,EQREL_1:23;
    hence x in R by A1,A16,EQREL_1:19;
  end;
  then
A18: R = [:g, g:]"(id the carrier of LR) by TARSKI:2;
  for x being Element of L holds g.x = Class(EqRel R, x) by A5;
  then g is CLHomomorphism of L, LR by A3;
  hence thesis by A18,Th34;
end;
