reserve N,M,K for ExtNat;
reserve X for ext-natural-membered set;

theorem Th56:
  for A, B being with_cardinal_domain Sequence st dom A in dom B
  holds A^B is with_cardinal_domain
proof
  let A, B be with_cardinal_domain Sequence;
  assume A1: dom A in dom B;
  A2: dom(A^B) = (dom A)+^(dom B) by ORDINAL4:def 1;
  per cases;
  suppose dom B is infinite;
    hence thesis by A1, A2, ThAdd;
  end;
  suppose A3: dom B is finite;
    dom A c= dom B by A1, ORDINAL1:def 2;
    hence thesis by A2, A3;
  end;
end;
