reserve x,y,z,X for set,
  T for Universe;

theorem Th1:
  for f being Function st dom f in T & rng f c= T holds product f in T
proof
  let f be Function such that
A1: dom f in T and
A2: rng f c= T;
  card dom f in card T by A1,CLASSES2:1;
  then card rng f in card T by CARD_2:61,ORDINAL1:12;
  then rng f in T by A2,CLASSES1:1;
  then union rng f in T by CLASSES2:59;
  then Union f in T by CARD_3:def 4;
  then product f c= Funcs(dom f, Union f) & Funcs(dom f, Union f) in T by A1,
CLASSES2:61,FUNCT_6:1;
  hence thesis by CLASSES1:def 1;
end;
