I need someone to explain this proof from James Munkres' Topology. 
The author writes $(X-C)\cap Y = Y-A,$ and, also, $A=Y \cap (X-U)$. I was wondering how is that something anyone writing an original proof of the theorem saw and if there is an analytic proof that the equalities holds true. Thanks for your help.

He defined $A = C \cap Y$; To see $(X-C) \cap Y = Y - A$, we can compute (as Frank did) or reason: a point is in the left hand side if it is not in $C$ but it is in $Y$. This means not in $A$ (to be in $A$, one has to be both in $C$ and $Y$), so it is in $Y - A$. On the other hand, if a point is in $Y - A$, it is in $Y$ but not in $A$, and the only way that could happen is if it is not in $C$ (otherwise it would be in $A$ by definition). So it is not in $C$ (so in $X-C$) and in $Y$, so in $Y \cap (X-C)$.
Or one could draw a schematic picture, and see it that way (it's quite obvious then).
