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 Th4:
  x in H1 * H2 iff ex a,b st x = a * b & a in H1 & b in H2
proof
  thus x in H1 * H2 implies ex a,b st x = a * b & a in H1 & b in H2
  proof
    assume x in H1 * H2;
    then x in carr H1 * H2;
    then consider a,b such that
A1: x = a * b and
A2: a in carr H1 and
A3: b in H2 by GROUP_2:94;
    a in H1 by A2,STRUCT_0:def 5;
    hence thesis by A1,A3;
  end;
  given a,b such that
A4: x = a * b & a in H1 and
A5: b in H2;
  b in carr H2 by A5,STRUCT_0:def 5;
  then x in H1 * carr H2 by A4,GROUP_2:95;
  hence thesis;
end;
