theorem
  x in A * H iff ex g1,g2 st x = g1 * g2 & g1 in A & g2 in H
proof
  thus x in A * H implies ex g1,g2 st x = g1 * g2 & g1 in A & g2 in H
  proof
    assume x in A * H;
    then consider g1,g2 such that
A1: x = g1 * g2 & g1 in A and
A2: g2 in carr H;
    g2 in H by A2;
    hence thesis by A1;
  end;
  given g1,g2 such that
A3: x = g1 * g2 & g1 in A and
A4: g2 in H;
  g2 in carr H by A4;
  hence thesis by A3;
end;
