reserve A for set, x,y,z for object,
  k for Element of NAT;
reserve n for Nat,
  x for object;
reserve V, C for set;

theorem Th22:
  for F being Function, X being set holds Card (F|X) = (Card F)|X
proof
  let F be Function, X be set;
A1: dom ((Card F)|X) = dom (Card F) /\ X by RELAT_1:61
    .= dom F /\ X by CARD_3:def 2
    .= dom (F|X) by RELAT_1:61;
  now
    let x be object;
A2: dom (F|X) c= dom F by RELAT_1:60;
    assume
A3: x in dom (F|X);
    hence ((Card F)|X).x = (Card F).x by A1,FUNCT_1:47
      .= card (F.x) by A3,A2,CARD_3:def 2
      .= card ((F|X).x) by A3,FUNCT_1:47;
  end;
  hence thesis by A1,CARD_3:def 2;
end;
