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 Th130
    .= 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;
