reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem Th42:
  [x,y] in (id X)*R iff x in X & [x,y] in R
proof
  thus [x,y] in (id X)*R implies x in X & [x,y] in R
  proof
    assume [x,y] in (id X)*R;
    then ex z st [x,z] in id X & [z,y] in R by Def6;
    hence thesis by Def8;
  end;
  assume x in X;
  then [x,x] in id X by Def8;
  hence thesis by Def6;
end;
