theorem BOOLE'67: :: BOOLE'67:

theorem BOOLE'67: :: BOOLE'67:
  (X /\ Y) /\ Z = X /\ (Y /\ Z)
proof
  thus (X /\ Y) /\ Z c= X /\ (Y /\ Z)
  proof let x;
    assume x in (X /\ Y) /\ Z;
    then x in X /\ Y & x in Z by XBOOLE_0:def 3;
    then x in X & x in Y & x in Z by XBOOLE_0:def 3;
    then x in X & x in Y /\ Z by XBOOLE_0:def 3;
    hence thesis by XBOOLE_0:def 3;
  end;
  let x;
  assume x in X /\ (Y /\ Z);
  then x in X & x in Y /\ Z by XBOOLE_0:def 3;
  then x in X & x in Y & x in Z by XBOOLE_0:def 3;
  then x in X /\ Y & x in Z by XBOOLE_0:def 3;
  hence thesis by XBOOLE_0:def 3;
end;

　これは以下の通りです。

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

証明　

$$\begin{eqnarray*}　 &&まず，(1)~~(X \cap Y) \cap Z \subseteq X \cap (Y \cap Z)... ...3から，\\ &&X \cap (Y \cap Z) \subseteq (X \cap Y) \cap Z\\ \end{eqnarray*}$$

よって，

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

証明終了