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 Th36:
  M is strongly_inaccessible implies Rank M is Tarski
proof
  assume
A1: M is strongly_inaccessible;
  thus X in Rank M & Y c= X implies Y in Rank M by CLASSES1:41;
  thus X in Rank M implies bool X in Rank M
  proof
    assume X in Rank M;
    then consider A such that
A2: A in M & X in Rank A by CLASSES1:31;
    succ A in M & bool X in Rank succ A by A2,CLASSES1:42,ORDINAL1:28;
    hence thesis by CLASSES1:31;
  end;
A3: cf M = M by A1,CARD_5:def 3;
  thus X c= Rank M implies X,Rank M are_equipotent or X in Rank M
  proof
A4: card X c< M iff card X c= M & card X <> M by XBOOLE_0:def 8;
    set R = the_rank_of X;
    assume that
A5: X c= Rank M and
A6: not X,Rank M are_equipotent and
A7: not X in Rank M;
    card X c= card Rank M & card X <> card Rank M by A5,A6,CARD_1:5,11;
    then
A8: card X in M by A1,A4,Th35,ORDINAL1:11;
    per cases;
    suppose
A9:   X = {};
      {} c= M;
      then {} c< M by XBOOLE_0:def 8;
      then
A10:  {} in M by ORDINAL1:11;
      M c= Rank M by CLASSES1:38;
      hence contradiction by A7,A9,A10;
    end;
    suppose
      X is non empty;
      then reconsider X1=X as non empty set;
      R in M
      proof
        deffunc f(set) = the_rank_of $1;
        set RANKS={ f(x) where x is Element of X1: x in X1};
A11:    for x st x in X holds x in Rank M by A5;
        RANKS c= M
        proof
          let y be object;
          assume y in RANKS;
          then consider x being Element of X1 such that
A12:      y = the_rank_of x and
          x in X1;
          x in Rank M by A11;
          hence thesis by A12,CLASSES1:66;
        end;
        then reconsider RANKS1=RANKS as Subset of M;
        ex N1 being Ordinal st (N1 in M & for x st x in X1 holds
        the_rank_of x in N1)
        proof
          assume
A13:      for N1 being Ordinal st N1 in M ex x st x in X1 & not
          the_rank_of x in N1;
          for N1 being Ordinal st N1 in M ex C st C in RANKS & N1 c= C
          proof
            let N1 be Ordinal;
            assume N1 in M;
            then consider x such that
A14:        x in X1 and
A15:        not the_rank_of x in N1 by A13;
            take C=the_rank_of x;
            thus C in RANKS by A14;
            thus thesis by A15,ORDINAL1:16;
          end;
          then RANKS1 is unbounded by Th6;
          then
A16:      cf M c= card RANKS1 by Th20;
          card RANKS c= card X1 from TREES_2:sch 2;
          then M c= card X1 by A3,A16;
          then card X1 in card X1 by A8;
          hence contradiction;
        end;
        then consider N1 being Ordinal such that
A17:    N1 in M and
A18:    for x st x in X1 holds the_rank_of x in N1;
        the_rank_of X c= N1 by A18,CLASSES1:69;
        hence thesis by A17,ORDINAL1:12;
      end;
      hence contradiction by A7,CLASSES1:66;
    end;
  end;
end;
