theorem
  rng g c= rng(f +* g)
proof
  let y be object;
  assume y in rng g;
  then consider x being object such that
A1: x in dom g & y = g.x by FUNCT_1:def 3;
  x in dom(f +* g) & (f +* g).x = y by A1,Th12,Th13;
  hence thesis by FUNCT_1:def 3;
end;
