theorem Th20:
  for M being Aleph for X being Subset of M holds X is unbounded
  implies cf M c= card X
proof
  let M be Aleph;
  let X be Subset of M;
  assume X is unbounded;
  then
A1: sup X = M;
  assume not cf M c= card X;
  then card X in cf M by ORDINAL1:16;
  then sup X in M by CARD_5:26;
  hence contradiction by A1;
end;
