reserve x,A,B,X,X9,Y,Y9,Z,V for set;

theorem Th111:
  (X /\ Z) \ (Y /\ Z) = (X \ Y) /\ Z
proof
  thus (X /\ Z) \ (Y /\ Z) = ((X /\ Z) \ Y) \/ ((X /\ Z) \ Z) by Lm2
    .= ((X /\ Z) \ Y) \/ (X /\ (Z \ Z)) by Th49
    .= ((X /\ Z) \ Y) \/ (X /\ {}) by Lm1
    .= (X \ Y) /\ Z by Th49;
end;
