reserve N for Cardinal;
reserve M for Aleph;
reserve X for non empty set;
reserve Y,Z,Z1,Z2,Y1,Y2,Y3,Y4 for Subset of X;
reserve S for Subset-Family of X;
reserve x for set;
reserve F,Uf for Filter of X;
reserve S for non empty Subset-Family of X;
reserve I for Ideal of X;
reserve S,S1 for Subset-Family of X;
reserve FS for non empty Subset of Filters(X);
reserve X for infinite set;
reserve Y,Y1,Y2,Z for Subset of X;
reserve F,Uf for Filter of X;
reserve x for Element of X;

theorem Th28:
  GCH implies (M is limit_cardinal implies M is strong_limit)
proof
  assume
A1: GCH;
  assume
A2: M is limit_cardinal;
  assume not M is strong_limit;
  then consider N being Cardinal such that
A3: N in M and
A4: not exp(2,N) in M;
A5: nextcard N c= M by A3,CARD_3:90;
A6: N is infinite
  proof
    assume N is finite;
    then Funcs(N,2) is finite by FRAENKEL:6;
    then card Funcs(N,2) is finite;
    then exp(2,N) is finite by CARD_2:def 3;
    hence thesis by A4,CARD_3:86;
  end;
  M c= exp(2,N) by A4,CARD_1:4;
  then M c= nextcard N by A1,A6;
  then nextcard N = M by A5;
  hence contradiction by A2;
end;
