theorem
  a1 [= b1 & a2 [= b2 implies a1 delta a2 [= b1 delta b2
proof
  assume a1 [= b1 & a2 [= b2;
  then Bottom b1 [= Bottom a1 & Bottom b2 [= Bottom a2 by Th13;
  then Bottom b1 [*] Bottom b2 [= Bottom a1 [*] Bottom a2 by Th9;
  hence thesis by Th13;
end;
