Use natural deduction to prove an easy logical statement

Question

How to prove $(A \rightarrow ( B\vee C) ) \rightarrow ((A \rightarrow B) \vee(A \rightarrow C))$ when $\vee$ means or, using natural deduction?

It is easy to prove the converse , but I didn't success to prove that.

Help me please. Thanks.

Answer 1

Hint: Use law of excluded middle to split cases cleverly. $\def\imp{\rightarrow}$

Sketch:

If $A \imp B \lor C$:

  $B \lor \neg B$. [Put the proof of LEM for $B$ before this.]

  If $B$:

    ...

    $A \imp B$.

    $( A \imp B ) \lor ( A \imp C )$.

  If $\neg B$:

    If $A$:

      $B \lor C$.

      ...

      $C$.

    $A \imp C$.

    $( A \imp B ) \lor ( A \imp C )$.

...