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: theorem :: SYSREL'2: : xboole1.miz : theorem BOOLE'30: :: BOOLE'30:   目次

theorem BOOLE'64: :: BOOLE'64: 

theorem BOOLE'64: :: BOOLE'64:
  (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 or x in Z by XBOOLE_0:def 2;
      then x in X or x in Y or x in Z by XBOOLE_0:def 2;
      then x in X or x in Y \/ Z by XBOOLE_0:def 2;
      hence thesis by XBOOLE_0:def 2;
   end;
  let x;
  assume x in X \/ (Y \/ Z);
  then x in X or x in Y \/ Z by XBOOLE_0:def 2;
  then x in X or x in Y or x in Z by XBOOLE_0:def 2;
  then x in X \/ Y or x in Z by XBOOLE_0:def 2;
  hence thesis by XBOOLE_0:def 2;
end;
　これは以下の通りです。


\begin{displaymath} (X \cup Y) \cup Z = X \cup (Y \cup Z) \end{displaymath}

証明

\begin{eqnarray*}　 &&まず，(1)~~(X \cup Y) \cup Z \subseteq X \cup (Y \cup Z)... ...蝓� \\ &&X \cup (Y \cup Z)　\subseteq　(X \cup Y) \cup Z \\ \end{eqnarray*}

よってこれと(1)から

\begin{displaymath} (X \cup Y) \cup Z = X \cup (Y \cup Z) \end{displaymath}

証明終了


Yasunari SHIDAMA