theorem
  dom P c= rng R implies rng(R*P) = rng P
proof
  assume
A1: for y being object st y in dom P holds y in rng R;
  thus rng(R*P) c= rng P by Th20;
  let z be object;
  assume z in rng P;
  then consider y being object such that
A2: [y,z] in P by XTUPLE_0:def 13;
  y in dom P by A2,XTUPLE_0:def 12;
  then y in rng R by A1;
  then consider x being object such that
A3: [x,y] in R by XTUPLE_0:def 13;
  [x,z] in R*P by A2,A3,Def6;
  hence thesis by XTUPLE_0:def 13;
end;
