theorem
  con_class a is finite or Left_Cosets Normalizer{a} is finite implies
  ex C being finite set st C = con_class a & card C = index Normalizer{a}
proof
A1: card con_class a = Index Normalizer {a} by Th132
    .= card Left_Cosets Normalizer {a};
  then
A2: con_class a,Left_Cosets Normalizer {a} are_equipotent by CARD_1:5;
  assume
A3: con_class a is finite or Left_Cosets Normalizer {a} is finite;
  then reconsider C = con_class a as finite set by A2,CARD_1:38;
  take C;
  thus C = con_class a;
  Left_Cosets Normalizer {a} is finite by A3,A2,CARD_1:38;
  hence thesis by A1,GROUP_2:def 18;
end;
