Suppose we're in a field F in which x,y,z are members. Consider the axiom of distribution which states that(adsbygoogle = window.adsbygoogle || []).push({});

xy + xz = x(y + z) for all x,y,z ∈ F.

Now consider this deduction:

xy + xz - xy ⟹ xy + x(z-y)

True no doubt, but to check my understanding aren't we missing a step? Shouldn't we more accurately say:

xy + xz - xy ⟹xy + xz + x(-y)⟹ xy + x(z-y)

I'm trying to self-study Rudin and just want to check my understanding.

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# Distributive Law in a Field

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