reserve n,k,b for Nat, i for Integer;

theorem Th4:
  for X,Y being finite natural-membered set st X misses Y
  holds dom (Sgm0(X)^Sgm0(Y)) = dom Sgm0(X\/Y)
  proof
    let X,Y be finite natural-membered set such that A1: X misses Y;
    thus dom (Sgm0(X)^Sgm0(Y)) = len (Sgm0(X)) + len (Sgm0(Y))
    by AFINSQ_1:def 3
    .= card (X) + len (Sgm0(Y)) by AFINSQ_2:20
    .= card (X) + card (Y) by AFINSQ_2:20
    .= card (X\/Y) by A1,CARD_2:40
    .= len Sgm0(X\/Y) by AFINSQ_2:20
    .= dom Sgm0(X\/Y);
  end;
