theorem Th47:
  rng R c= Y implies R*(id Y) = R
proof
  assume
A1: rng R c= Y;
  R c= R*(id Y)
  proof
    let x,y;
    assume
A2: [x,y] in R;
    then y in rng R by XTUPLE_0:def 13;
    then [y,y] in (id Y) by A1,Def8;
    hence thesis by A2,Def6;
  end;
  hence thesis by Th44;
end;
