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 c= dom (S|X) implies R*(S|X) = R*S
proof
  assume
A1: rng R c= dom (S|X);
  let a,b;
  R*(S|X) c= R*S by Th23,Th53;
  hence [a,b] in R*(S|X) implies [a,b] in R*S;
  assume [a,b] in R*S;
  then consider c such that
A2: [a,c] in R and
A3: [c,b] in S by Def6;
  c in rng R by A2,XTUPLE_0:def 13;
  then c in X by A1,Th51;
  then [c,b] in S|X by A3,Def9;
  hence thesis by A2,Def6;
end;
