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
  id X /\ EqR = id X
proof
  now
    let x,y be object;
    assume [x,y] in id X;
    then x in X & x = y by RELAT_1:def 10;
    hence [x,y] in EqR by Th5;
  end;
  then id X c= EqR;
  hence thesis by XBOOLE_1:28;
end;
