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Now then suppose X + Y = Y + X is true when
Y has a particular value say N. That is
suppose X + N = N + X. Then it will
also be true when Y = 1 + N. That is
X + (1 + N) = (1 + N) + X. For
X + (1 + N) = (X + 1) + N and
X + 1 = 1 + X as we have just proved
so that
X + (1 + N) = (X + 1) + N = (1 + X) + N = 1 + (X + N).
But we are going on the supposition that X + N = N + X
so that X + (1 + N) = 1 + (N + X) = (1 + N) + X.
So the proposition X + Y = Y + X if true when Y has any particular
value is true when it has the next greater value.
But it is true when Y = 1 whatever be the value of X. Hence it must be
true for all values of X and Y.

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