A simple proof involving Peano's axioms

Hello all,
I was going through these integer axioms.



Try this:http://www.pitt.edu/~dickinsm/1020-2071/integerAxioms.pdf


1) I want to prove, -(-a) = a.

Is it possible to prove it without resorting to induction axiom?.
Looks to me like you can't.
In the particilar pdf above, the author claims it should be possible
to prove a=-(-a), without using the induction axiom. (look at
exercise
1).
I am wondering how that is possible.


On a similar line, how do we prove :
2) for any x,y,zEN, (x+z = y+z) ==> x=y.


Can this be proved without using the induction axiom?.


this one seems such a basic principle of addition that I am
'disappointed' that we have to use the induction axiom.
I thought may be it should just 'fall out' from the definition of
addition.
(However, the only definition of addition in Peano like systems seems
to be :
for any x,y (x + succ(y) ) = succ(x+y). )


If any of you feel I dont need induction axiom to prove the above two
statements please advice.
Thanks
PS: By induction axiom I mean the one that says - { If a statement P
is true for 0 and (if it is true for any nEN implies it is true for
succ(n) ) } ==> P is true for all N.



Thanks
.

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