theorem
  a * (H1 /\ H2) = (a * H1) /\ (a * H2)
proof
  thus a * (H1 /\ H2) c= (a * H1) /\ (a * H2)
  proof
    let x be object;
    assume x in a * (H1 /\ H2);
    then consider g such that
A1: x = a * g and
A2: g in H1 /\ H2 by Th103;
    g in H2 by A2,Th82;
    then
A3: x in a * H2 by A1,Th103;
    g in H1 by A2,Th82;
    then x in a * H1 by A1,Th103;
    hence thesis by A3,XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A4: x in (a * H1) /\ (a * H2);
  then x in a * H1 by XBOOLE_0:def 4;
  then consider g such that
A5: x = a * g and
A6: g in H1 by Th103;
  x in a * H2 by A4,XBOOLE_0:def 4;
  then consider g1 such that
A7: x = a * g1 and
A8: g1 in H2 by Th103;
  g = g1 by A5,A7,GROUP_1:6;
  then g in H1 /\ H2 by A6,A8,Th82;
  hence thesis by A5,Th103;
end;
