reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;

theorem Th23:
  for x st x in X holds y in Class(EqR,x) iff Class(EqR,x) = Class (EqR,y)
proof
  let x such that
A1: x in X;
  thus y in Class(EqR,x) implies Class(EqR,x) = Class(EqR,y)
  proof
    assume y in Class(EqR,x);
    then [y,x] in EqR by Th19;
    then [x,y] in EqR by Th6;
    hence thesis by A1,Lm2;
  end;
  assume Class(EqR,x) = Class(EqR,y);
  then [x,y] in EqR by A1,Lm2;
  then [y,x] in EqR by Th6;
  hence thesis by Th19;
end;
