
theorem
  :: Corollary 2.13, p. 62, (congruence --> kernel --> congruence)
  for L being continuous complete LATTICE, R being non empty Subset of
[:L, L :] st R is CLCongruence for x being set holds x is Element of L./.R iff
  ex y being Element of L st x = Class(EqRel R, y)
proof
  let L be continuous complete LATTICE, R be non empty Subset of [:L, L:];
  assume R is CLCongruence;
  then
A1: the carrier of (L./.R) = Class EqRel R by Def5;
  let x be set;
  hereby
    assume x is Element of L./.R;
    then x in Class EqRel R by A1;
    then consider y being object such that
A2: y in the carrier of L and
A3: x = Class(EqRel R, y) by EQREL_1:def 3;
    reconsider y as Element of L by A2;
    take y;
    thus x = Class(EqRel R, y) by A3;
  end;
  given y being Element of L such that
A4: x = Class(EqRel R, y);
  thus thesis by A1,A4,EQREL_1:def 3;
end;
