
theorem Th8:
  for f, g, h being Function st rng f c= dom h holds (g +* h)*f = h*f
proof
  let f, g, h be Function;
  assume A1: rng f c= dom h;
  then rng f c= dom g \/ dom h by XBOOLE_1:10;
  then rng f c= dom(g +* h) by FUNCT_4:def 1;
  then A2: dom((g +* h)*f) = dom f by RELAT_1:27
    .= dom(h*f) by A1, RELAT_1:27;
  now
    let x be object;
    assume A3: x in dom((g +* h)*f);
    then A4: x in dom f by FUNCT_1:11;
    then A5: f.x in rng f by FUNCT_1:3;
    thus ((g +* h)*f).x = (g +* h).(f.x) by A3, FUNCT_1:12
      .= h.(f.x) by A1, A5, FUNCT_4:13
      .= (h*f).x by A4, FUNCT_1:13;
  end;
  hence thesis by A2, FUNCT_1:2;
end;
