Isn't this just a property of equality?
This is trivial. But I could not work out a formal prove for this (using the axioms of addition).
Seems like we have to rely of definition rather than a formal prove.
Let us for the moment assume
A=B => B-A = 0 Let is call it (1)
A+C = A + C + B - A = B + C
But to prove (1) above we somewhere have to rely on A=B => A+C = B+C
Any help please?
Yes - Makes sense. Though I had to prove to myself the following result
"=> A+C > B+C or A+C < B+C
=> A > B or A < B (by definitions of inequalities)"
You can also prove the original from the definition of mapping.
If x=y then f(x) = f(y), as others suggested.
Doing that, you assume your field is totally ordered, which is often wrong.Lets assume ,
A=B => A+C != B+C (!= stands for not equal)
=> A+C > B+C or A+C < B+C
=> A > B or A < B (by definitions of inequalities)
The answer is just the fact that if A=B, then any sentence in which you replace A by B or B by A will keep its meaning ("a property of equality").