Neither is a matter of proof but of definition.
There is a great convenience in having these properties because then the general formulas that work when all quanitties are positive remain valid when some or all of them are allowed to be negative.
(a + b)c = ab + ac
Allowing for a negative, say, b
0 = (a - a)c = ac - ac = 0
fits nicely. But, again, the latter is a matter of definition, not of a derivation.