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i start to self learn calculus tom apostol (in my college we got another book)

i start from the very begining of the chapter. so this not really a homework. if i post thread in wrong sub-forum please remind me :)

## Homework Statement

prove : If a+b = a+c, then b = c.

## Homework Equations

this also prove zero number is unique.

## The Attempt at a Solution

the prove was easy

using AXIOM EXISTENCE OF IDENTITY ELEMENTS & ASSOCIATIVE

there is a number y such that y + a = O

then

y+(a+b) = y+(a+c)

(y+a)+b=(y+a)+c

0+b=0+c

b=c

but why in the book say this also prove number 0 is unique? in other words only one real number have the property zero?!

thank u :)