theorem Th63:
  (H1 /\ H2) * a = (H1 * a) /\ (H2 * a)
proof
  let g;
  thus g in (H1 /\ H2) * a implies g in (H1 * a) /\ (H2 * a)
  proof
    assume g in (H1 /\ H2) * a;
    then consider h such that
A1: g = h * a and
A2: h in H1 /\ H2 by Th58;
    h in H2 by A2,Th82;
    then
A3: g in H2 * a by A1,Th58;
    h in H1 by A2,Th82;
    then g in H1 * a by A1,Th58;
    hence thesis by A3,Th82;
  end;
  assume
A4: g in (H1 * a) /\ (H2 * a);
  then g in H1 * a by Th82;
  then consider b such that
A5: g = b * a and
A6: b in H1 by Th58;
  g in H2 * a by A4,Th82;
  then consider c such that
A7: g = c * a and
A8: c in H2 by Th58;
  b = c by A5,A7,ThB16;
  then b in (H1 /\ H2) by A6,A8,Th82;
  hence thesis by A5,Th58;
end;
