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Proof of 4.1
By definition of regularity of $f$ and $f|_{\partial M}$, $T_x f((\partial M)_x \cap (f^{-1}(y))_x)$ = $T_x f((\partial M)_x)$ $\cap$ $T_x f(f^{-1}(y))_f(x)$ = $(f|_{\partial M}(\partial M))_x$ $\cap$ ${y}_{f(x)}$ = ${{y}_{f(x)}}$ = $T_x f((f^{-1}(y))_x)$.
So, $T_x f((f^{-1}(y)_x)$ $...
Is it legally acceptable for an active service member to willfully abandoned his family members (wife and 2 young children) and then a year and a half later, accept an honorable seperation with a party, a paddle and no communication between the family and the CO?
