reserve A,B for limit_ordinal infinite Ordinal;
reserve B1,B2,B3,B5,B6,D, C for Ordinal;
reserve X for set;
reserve X for Subset of A;
reserve M for non countable Aleph;
reserve X for Subset of M;
reserve N,N1 for cardinal infinite Element of M;
reserve A for Ordinal;
reserve x,y,X,Y for set;

theorem Th33:
  for M being Aleph holds (card X in cf M & for Y st Y in X holds
  card Y in M) implies card union X in M
proof
  let M be Aleph;
  assume
A1: card X in cf M;
  assume
A2: for Y st Y in X holds card Y in M;
  per cases;
  suppose
A3: X = {};
    {} c= M;
    then {} c< M by XBOOLE_0:def 8;
    hence thesis by A3,ORDINAL1:11,ZFMISC_1:2;
  end;
  suppose
    X is non empty;
    then reconsider X1=X as non empty set;
    deffunc f(set) = card $1;
    set CARDS = { f(Y) where Y is Element of X1 : Y in X1 };
    card CARDS c= card X1 from TREES_2:sch 2;
    then
A4: card CARDS in cf M by A1,ORDINAL1:12;
A5: for x st x in CARDS holds x in M & ex y st y in CARDS & x c= y & y is
    Cardinal
    proof
      let x;
      assume x in CARDS;
      then consider Y being Element of X1 such that
A6:   card Y = x and
      Y in X1;
      thus x in M by A2,A6;
      take card Y;
      thus thesis by A6;
    end;
    then
    for x st x in CARDS holds ex y st y in CARDS & x c= y & y is Cardinal;
    then reconsider UN=union CARDS as Cardinal by Th32;
    for x being object st x in CARDS holds x in M by A5;
    then CARDS c= M;
    then UN in M by A4,CARD_5:26;
    then
A7: (card X1) *` UN in cf M *` M by A1,CARD_4:20;
    for Y st Y in X1 holds card Y c= UN
    proof
      let Y;
      assume Y in X1;
      then card Y in CARDS;
      hence thesis by ZFMISC_1:74;
    end;
    then
A8: card union X1 c= (card X1) *` UN by CARD_2:87;
    cf M c= M by CARD_5:def 1;
    then (cf M) *` M c= M *` M by CARD_2:90;
    then (cf M) *` M c= M by CARD_4:15;
    hence thesis by A8,A7,ORDINAL1:12;
  end;
end;
