reserve x,y for set,
  k,n for Nat,
  i for Integer,
  G for Group,
  a,b,c ,d,e for Element of G,
  A,B,C,D for Subset of G,
  H,H1,H2,H3,H4 for Subgroup of G ,
  N1,N2 for normal Subgroup of G,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem
  a in H & b in H implies a |^ b in H
proof
  assume a in H & b in H;
  then b" in H & a * b in H by GROUP_2:50,51;
  then b " * (a * b) in H by GROUP_2:50;
  hence thesis by GROUP_1:def 3;
end;
