Many times in mathematics in proving something we can put
x=a+b+c or x= a-b^2 or x=y e.t.c.
What axiom or theorem in mathematics allow us to do that??
in the following example how the function concept could be used?
suppose we want to prove that:
...................(a+b+c)/3 >= (abc)^1/3,with a,b,c>=0
Now without using the Am-Gm concept for the proof of the inequality,
one way of solving the inequality is to use the substitution
which is the desired result (also see the proof for Young's inequality).
As ThePerfectHacker mentioned, there is no axiom on the things you told.
You just find your way yourself by picking new parameters or arranging the given ones to reveal your own way.
Ofcourse, experience is the most important thing in this dark forest!
In my very 1st post i asked:
......what allows us to put x=a+b+c ...............................................
in a proof that uses this substitution.And then i brought up as an example to the posts of the PerfectHacker the said inequality
Anyway in your proof you do not use ,if i am not mistaken,any substitutions
PerfectHacker brought in the concept of function and i was forced to point out that even the concept of function needs a theorem of existence.
So now we are faced with two questions ;
1)If the concept of function needs a theorem of existence,or
2)The substitution x=a+b+c, or any substitution of this type need a theorem or axiom to back it up
For example the concept of sqroot need a theorem of existence before it is defined
Notice, my ' or" is not exclusively disjunctive
However, if a proof depends on the assumption it is not a proof. You need to prove the assumption.
In the example I gave you there is nothing bad. I was not trying to prove anything. Just give you an example. Everything was justified.