reserve I,X,x,d,i for set;
reserve M for ManySortedSet of I;
reserve EqR1,EqR2 for Equivalence_Relation of X;
reserve I for non empty set;
reserve M for ManySortedSet of I;
reserve EqR,EqR1,EqR2,EqR3,EqR4 for Equivalence_Relation of M;
reserve S for non void non empty ManySortedSign;
reserve A for non-empty MSAlgebra over S;

theorem
  for X be set holds x in the carrier of EqRelLatt X iff x is
  Equivalence_Relation of X
proof
  let X be set;
A1: the carrier of EqRelLatt X = { x1 where x1 is Relation of X,X : x1 is
  Equivalence_Relation of X } by Def2;
  thus x in the carrier of EqRelLatt X implies x is Equivalence_Relation of X
  proof
    assume x in the carrier of EqRelLatt X;
    then
    ex x1 be Relation of X,X st x = x1 & x1 is Equivalence_Relation of X by A1;
    hence thesis;
  end;
  thus thesis by A1;
end;
