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
  for H being strict Subgroup of G holds H |^ a = (1).G implies H = (1). G
proof
  let H be strict Subgroup of G;
  assume
A1: H |^ a = (1).G;
  then reconsider H as finite Subgroup of G by Th65;
  card (1).G = 1 by GROUP_2:69;
  then card H = 1 by A1,Th64;
  hence thesis by GROUP_2:70;
end;
