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 Th26:
  M is strongly_Mahlo implies M is Mahlo
  proof
    assume M is strongly_Mahlo; then
A2: { N : N is strongly_inaccessible} is_stationary_in M;
A3: { N : N is strongly_inaccessible} c= { N1 : N1 is regular }
    proof
      let x be object;
      assume x in { N : N is strongly_inaccessible}; then
      consider N such that B1: x = N & N is strongly_inaccessible;
      x in { N1 : N1 is regular } by B1;
      hence thesis;
    end;
    { N : N is regular } c= M
    proof
      let x be object;
      assume x in { N : N is regular }; then
      consider N such that A1: x = N & N is regular;
      thus x in M by A1;
    end; then
    { N : N is regular } is_stationary_in M by A2,A3,Th14; then
    M is Mahlo;
    hence thesis;
  end;
