Proof reasoning

  1. Explain why it is necessary to prove both ``if TeX Embedding failed! then TeX Embedding failed!" and ``if TeX Embedding failed! then TeX Embedding failed!" in order to prove ``TeX Embedding failed! if and only if TeX Embedding failed!".

    In order to prove ``TeX Embedding failed! if and only if TeX Embedding failed!" (aka ``TeX Embedding failed! iff TeX Embedding failed!") one must prove both ``if TeX Embedding failed! then TeX Embedding failed!" and ``if TeX Embedding failed! then TeX Embedding failed!". But why both? The necessity of ``if TeX Embedding failed! then TeX Embedding failed!" is obvious from the statement of the question and corresponds to the ``if" part. The ``only if" part refers to the notion that TeX Embedding failed! is true only when TeX Embedding failed! is true, i.e that TeX Embedding failed! is not true when TeX Embedding failed! is false, this is proven by showing that whenever TeX Embedding failed! is true, TeX Embedding failed! is also true and hence that TeX Embedding failed! is never false when TeX Embedding failed! is true, hence the expression ``only if".