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 Th73:
  G is commutative Group implies for H being strict Subgroup of G
  for a holds H |^ a = H
proof
  assume
A1: G is commutative Group;
  let H be strict Subgroup of G;
  let a;
  the carrier of H |^ a = a" * H * a by Th59
    .= H * a" * a by A1,GROUP_2:112
    .= H * (a" * a) by GROUP_2:107
    .= H * 1_G by GROUP_1:def 5
    .= the carrier of H by GROUP_2:109;
  hence thesis by GROUP_2:59;
end;
