#9 (X - Y) - (Z -W) = (X + W) - Y -Z

First: (X - Y) - (Z -W) = (X - Y) + ( -Z + W )

Uses: Associative & Distributive Laws and, repeatedly, that P + (-1)Q = P - Q (call that fact *).

(X - Y) - (Z -W)

= (X - Y) + (-1)(Z + (-1)W)............................(*) and (*)

= (X - Y) + ( (-1)(Z) + (-1)[(-1)(W)] ).............Distributive Law

= (X - Y) + ( -Z + [(-1)(-1)](W) )....................(*) and Associative Law (for Multiplication)

= (X - Y) + ( -Z + [1](W) ).............................Arithmetic: (-1)(-1) = 1

= (X - Y) + ( -Z + W )....................................1 is the unit for multiplcation

Second: (X - Y) + ( -Z + W ) = (X + W) - Y -Z

Uses: The above, the Laws, and the identity: -( Y + Z ) = -Y - Z

(X - Y) + ( -Z + W )

= (X - Y) + ( W - Z ).......................................Commutativ e Law (for addition)

= (X + (-1)Y) + ( W + (-1)Z )..........................(*) and (*)

= [ (X + (-1)Y) + (W) ] + (-1)Z........................Associative Law (for addition), treating (X + (-1)Y) as a single value.

= [ X + ( (-1)Y + W ) ] + (-1)Z........................Associative Law (for addition).

= [ X + ( W + (-1)Y ) ] + (-1)Z........................Commutative Law (for addition).

= [ ( X + W ) + ( (-1)Y ) ] + (-1)Z....................Associative Law (for addition).

= ( X + W ) + [ (-1)Y + (-1)Z ]........................Associative Law (for addition), treating (X+W) as a single value.

= ( X + W ) + (-1)[ Y + Z ].............................Distributive Law.

= ( X + W ) - ( Y + Z )....................................(*)

= ( X + W ) - Y - Z.........................................Given identity.