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

theorem
  X meets Y & X misses Z implies X meets Y \ Z
proof
  assume that
A1: X meets Y and
A2: X misses Z;
  X /\ (Y \ Z) = X /\ Y \ X /\ Z by Th50
    .= X /\ Y \ {} by A2;
  hence X /\ (Y \ Z) <> {} by A1;
end;
