theorem Th38:
  for h being Function of Y,X holds (for z being Element of Y
  holds h.z = F.(f.z,g.z)) implies h = F.:(f,g)
proof
  let h be Function of Y,X;
  assume
A1: for z being Element of Y holds h.z = F.(f.z,g.z);
  now
    let z be Element of Y;
    thus h.z = F.(f.z,g.z) by A1
      .= (F.:(f,g)).z by Th37;
  end;
  hence thesis by FUNCT_2:63;
end;
