reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;
reserve L for Subset of Subgroups G;
reserve N2 for normal Subgroup of G;

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;
