
theorem Th3:
  for f1,f2,g being Function st rng g c= dom f2 holds (f1+*f2)*g = f2*g
proof
  let f1,f2,g be Function;
  assume
A1: rng g c= dom f2;
A2: now
    let x be object;
    assume
A3: x in dom g;
    then
A4: (f2*g).x = f2.(g.x) by FUNCT_1:13;
A5: g.x in rng g by A3,FUNCT_1:3;
    ((f1+*f2)*g).x = (f1+*f2).(g.x) by A3,FUNCT_1:13;
    hence ((f1+*f2)*g).x = (f2*g).x by A1,A4,A5,FUNCT_4:13;
  end;
  dom (f1+*f2) = dom f1 \/ dom f2 by FUNCT_4:def 1;
  then
A6: dom ((f1+*f2)*g) = dom g by A1,RELAT_1:27,XBOOLE_1:10;
  dom (f2*g) = dom g by A1,RELAT_1:27;
  hence thesis by A6,A2;
end;
