Suppose a,b e G where G is a group under addition.

Then, the equation x + a = b has a unique solution in G. The solution is

x = b - a.

Proof:

Suppose the equation x + a = b has two solutions. Let the solutions be

x = b - a and y = b - a.

Now, by transitivity of equality, x = y. THus there is a unique solution in G.

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