
theorem Th35:
  for L being complete LATTICE, R being non empty Subset of [:L, L
:] st R is CLCongruence for x be Element of L holds [inf Class(EqRel R, x), x]
  in R
proof
  let L be complete LATTICE, R be non empty Subset of [:L, L:];
  assume
A1: R is CLCongruence;
  let x be Element of L;
  set CRx = Class(EqRel R, x);
  reconsider SR = [:CRx, {x}:] as Subset of [:L, L:];
  R is Equivalence_Relation of the carrier of L by A1;
  then
A2: R = EqRel R by Def1;
  SR c= the carrier of subrelstr R
  proof
    let a be object;
    assume a in SR;
    then consider a1, a2 being object such that
A3: a1 in CRx and
A4: a2 in {x} and
A5: a = [a1, a2] by ZFMISC_1:def 2;
    a2 = x by A4,TARSKI:def 1;
    then a in R by A2,A3,A5,EQREL_1:19;
    hence thesis by YELLOW_0:def 15;
  end;
  then reconsider SR9 = SR as Subset of subrelstr R;
A6: ex_inf_of SR, [:L, L:] by YELLOW_0:17;
  subrelstr R is CLSubFrame of [:L, L:] by A1;
  then
A7: "/\"(SR9, [:L, L:]) in the carrier of subrelstr R by A6,YELLOW_0:def 18;
A8: x in CRx by EQREL_1:20;
  inf SR = [inf proj1 SR, inf proj2 SR] by Th7,YELLOW_0:17
    .= [inf CRx, inf proj2 SR] by FUNCT_5:9
    .= [inf CRx, inf {x}] by A8,FUNCT_5:9
    .= [inf CRx, x] by YELLOW_0:39;
  hence thesis by A7,YELLOW_0:def 15;
end;
