  After recalling briefly the connection between spontaneous symmetry breaking
and off-diagonal long range order for models of magnets a general proof of
spontaneous breaking of gauge symmetry as a consequence of Bose-Einstein
Condensation is presented. The proof is based on a rigorous validation of
Bogoliubov's $c$-number substitution for the ${\bf k}={\bf 0}$ mode operator
$a_{\bf 0}$.
