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 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;
