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
  G is commutative Group implies (x in H1 "\/" H2 iff ex a,b st x = a *
  b & a in H1 & b in H2)
proof
  assume G is commutative Group;
  then H1 * H2 = H2 * H1 by GROUP_2:25;
  hence thesis by Th5;
end;
