theorem
  for F,f,g being Function st [:rng f, rng g:] c= dom F holds dom(F.:(f,
  g)) = dom f /\ dom g
proof
  let F,f,g be Function such that
A1: [:rng f, rng g:] c= dom F;
  rng<:f,g:> c= [:rng f, rng g:] by FUNCT_3:51;
  hence dom(F.:(f,g)) = dom<:f,g:> by A1,RELAT_1:27,XBOOLE_1:1
    .= dom f /\ dom g by FUNCT_3:def 7;
end;
