theorem Th23:
  omega c= cf a
proof
A1: a is_cofinal_with cf a by Def1;
  then cf a <> {} by ORDINAL2:50;
  then
A2: {} in cf a by ORDINAL3:8;
  cf a is limit_ordinal by A1,ORDINAL4:38;
  hence thesis by A2,ORDINAL1:def 11;
end;
