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;

theorem Th30:
  M is strongly_Mahlo implies M is strong_limit
proof
  assume M is strongly_Mahlo;
  then
A1: { N : N is strongly_inaccessible} is_stationary_in M;
  then reconsider SI={N : N is strongly_inaccessible} as Subset of M;
  assume not M is strong_limit;
  then consider M1 being Cardinal such that
A2: M1 in M and
A3: not exp(2,M1) in M by CARD_FIL:def 14;
  succ M1 in M by A2,ORDINAL1:28;
  then M \ succ M1 is closed unbounded by Th12;
  then SI /\ (M \ succ M1) <> {} by A1;
  then consider M2 being object such that
A4: M2 in SI /\ (M \ succ M1) by XBOOLE_0:def 1;
  M2 in SI by A4,XBOOLE_0:def 4;
  then consider N such that
A5: N = M2 and
A6: N is strongly_inaccessible;
  M2 in (M \ succ M1) by A4,XBOOLE_0:def 4;
  then not N in succ M1 by A5,XBOOLE_0:def 5;
  then not N c= M1 by ORDINAL1:22;
  then M1 in N by ORDINAL1:16;
  then exp(2,M1) in N by A6,CARD_FIL:def 14;
  hence contradiction by A3,ORDINAL1:10;
end;
