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 P implies dom(R*P) = dom R
proof
  assume
A1: for y being object holds y in rng R implies y in dom P;
  thus dom(R*P) c= dom R by Th19;
  let x be object;
  assume x in dom R;
  then consider y being object such that
A2: [x,y] in R by XTUPLE_0:def 12;
  y in rng R by A2,XTUPLE_0:def 13;
  then y in dom P by A1;
  then consider z being object such that
A3: [y,z] in P by XTUPLE_0:def 12;
  [x,z] in R*P by A2,A3,Def6;
  hence thesis by XTUPLE_0:def 12;
end;
