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Now here is where I might be naive. I think that one direct way to prove that expression "c" is not equal to expression "b" is to simply subtract the two expressions. That is, I assert that expression "b" is equal to expression "c" if and only if you get zero when you subtract them. If not, then they are not equal.

In fact, what if you get the equality expression, after subtracting "b" and "c", an expression that reads b = c.

Well, I think this is circular reasoning. The answer gives what we are trying to prove that which we assume to be true. Is this a proof by contradiction that expression "c" cannot be equal to expression "a"?