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

theorem ThAdd:
  for A being Ordinal, B being infinite Cardinal st A in B holds A +^ B = B
proof
  let A being Ordinal, B be infinite Cardinal;
  assume A1: A in B;
  deffunc F(Ordinal) = A+^$1;
  consider fi being Ordinal-Sequence such that
    A2: dom fi = B & for C being Ordinal st C in B holds fi.C = F(C)
    from ORDINAL2:sch 3;
  A3: A +^ B = sup fi by A2, ORDINAL2:29;
  now
    let D be Ordinal;
    assume D in rng fi;
    then consider x being object such that
      B1: x in dom fi & fi.x = D by FUNCT_1:def 3;
    reconsider C = x as Ordinal by B1;
    card A in B & card C in B by A1, A2, B1, CARD_1:9;
    then card A +` card C in B +` B by CARD_2:96;
    then B2: card A +` card C in card B by CARD_2:75;
    D = A +^ C by A2, B1;
    then card D = card A +` card C by CARD_2:13;
    hence D in B by B2, CARD_3:43;
  end;
  then sup rng fi c= B by ORDINAL2:20;
  then A4: sup fi c= B by ORDINAL2:def 5;
  B c= A +^ B by ORDINAL3:24;
  hence thesis by A3, A4, XBOOLE_0:def 10;
end;
