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;

theorem Th65:
  H is finite iff H |^ a is finite
proof
  card H = card(H |^ a) by Th64;
  then the carrier of H,the carrier of H |^ a are_equipotent by CARD_1:5;
  hence thesis by CARD_1:38;
end;
