reserve X for non empty set;

theorem Th4:
  for A, x being set holds x in the carrier of EqRelLATT A iff x is
  Equivalence_Relation of A
proof
  let A, x be set;
  hereby
    assume x in the carrier of EqRelLATT A;
    then reconsider e = x as Element of LattPOSet EqRelLatt A;
    %e = e;
    then
A1: x in the carrier of EqRelLatt A;
    the carrier of EqRelLatt A = {r where r is Relation of A,A : r is
    Equivalence_Relation of A} by MSUALG_5:def 2;
    then
    ex x9 being Relation of A,A st x9 = x & x9 is Equivalence_Relation of
    A by A1;
    hence x is Equivalence_Relation of A;
  end;
A2: the carrier of EqRelLatt A = {r where r is Relation of A,A : r is
  Equivalence_Relation of A} by MSUALG_5:def 2;
  assume x is Equivalence_Relation of A;
  then x in the carrier of EqRelLatt A by A2;
  then reconsider e = x as Element of EqRelLatt A;
  reconsider e as Element of EqRelLATT A;
  e in the carrier of EqRelLATT A;
  hence thesis;
end;
