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
  rng R = rng Q implies rng(R*S) = rng (Q*S)
proof
  assume
A1: rng R = rng Q;
  thus rng(R*S) = S.:rng R by Th119
    .= rng (Q*S) by A1,Th119;
end;
