theorem
  (H1 /\ H2) * a = (H1 * a) /\ (H2 * a)
proof
  thus (H1 /\ H2) * a c= (H1 * a) /\ (H2 * a)
  proof
    let x be object;
    assume x in (H1 /\ H2) * a;
    then consider g such that
A1: x = g * a and
A2: g in H1 /\ H2 by Th104;
    g in H2 by A2,Th82;
    then
A3: x in H2 * a by A1,Th104;
    g in H1 by A2,Th82;
    then x in H1 * a by A1,Th104;
    hence thesis by A3,XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A4: x in (H1 * a) /\ (H2 * a);
  then x in H1 * a by XBOOLE_0:def 4;
  then consider g such that
A5: x = g * a and
A6: g in H1 by Th104;
  x in H2 * a by A4,XBOOLE_0:def 4;
  then consider g1 such that
A7: x = g1 * a and
A8: g1 in H2 by Th104;
  g = g1 by A5,A7,GROUP_1:6;
  then g in H1 /\ H2 by A6,A8,Th82;
  hence thesis by A5,Th104;
end;
