theorem BOOLE'86:

BOOLE'86: X \ (Y /\ Z) = (X \ Y) \/ (X \ Z)
proof
  thus X \ (Y /\ Z) c= (X \ Y) \/ (X \ Z)
  proof
    let x;
    assume x in X \ (Y /\ Z);
    then x in X & not x in (Y /\ Z) by XBOOLE_0:def 4;
    then x in X & (not x in Y or not x in Z) by XBOOLE_0:def 3;
    then x in (X \ Y) or x in (X \ Z) by XBOOLE_0:def 4;
    hence thesis by XBOOLE_0:def 2;
  end;
  Y /\ Z c= Y & Y /\ Z c= Z by BOOLE'37;
  then (X \ Y) c= X \ (Y /\ Z) & X \ Z c= X \ (Y /\ Z) by BOOLE'47;
  hence (X \ Y) \/ (X \ Z) c= X \ (Y /\ Z) by BOOLE'32;
end;

　これは以下の通りです。

$$X \setminus (Y \cap Z) = (X \setminus Y) \cup (X \setminus Z)$$

証明　

$$\begin{eqnarray*}　 && X \setminus (Y \cap Z) \subseteq (X \setminus Y) \cu... ...us Y) \cup (X \setminus Z) \subseteq X \setminus (Y \cap Z) \\ \end{eqnarray*}$$

よって，

$$X \setminus (Y \cap Z) = (X \setminus Y) \cup (X \setminus Z)$$

証明終了