
theorem Th1:
  for f,g,h being Function
  for A being set st A c= dom f & A c= dom g & rng h c= A &
  for x being set st x in A holds f.x = g.x holds f*h = g*h
proof
  let f,g,h be Function;
  let A be set such that
A1: A c= dom f and
A2: A c= dom g and
A3: rng h c= A and
A4: for x being set st x in A holds f.x = g.x;
A5: dom (f*h) = dom h by A1,A3,RELAT_1:27,XBOOLE_1:1;
A6: dom (g*h) = dom h by A2,A3,RELAT_1:27,XBOOLE_1:1;
  now
    let x be object;
    assume
A7: x in dom h;
    then
A8: (f*h).x = f.(h.x) by FUNCT_1:13;
A9: (g*h).x = g.(h.x) by A7,FUNCT_1:13;
    h.x in rng h by A7,FUNCT_1:3;
    hence (f*h).x = (g*h).x by A3,A4,A8,A9;
  end;
  hence thesis by A5,A6;
end;
