if a>0 and b & b<0 then |a|- |b| = ?
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Is it a mathematical proof that if A=0 and B=0, then A=B?
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2 Answers

Darryl Nester, 25+ years teaching college math
Answered January 13, 2017
Is it a mathematical proof that if A=0 and B=0, then A=B?
The transitive property of equality has been mentioned in other answers.
Here is another (more technical) way to answer this question: How do we know that there is only one number which behaves the way that “0” does?
When defining groups, fields, and rings using axioms, we often start with a set S, and a binary operation (say, addition) on that set. Then we state (in an axiom) that the set in question has an additive identity — that is, there exists an element e∈Se∈S with the behavior that, for any x, e+x=x+e=xe+x=x+e=x. The axiom does not specify that this object is unique, but typically it is followed by:
Theorem: The additive identity is unique.
Proof: Suppose ee and e′e′ both have the behavior of an additive identity. Then
e=e+e′=e′(□)(◻)e=e+e′=e′
To be clear: e=e+e′e=e+e′ because e′e′ behaves like the additive identity, and e+e′=e′e+e′=e′ because ee behaves like the additive identity. The proof of the uniqueness of the multiplicative identity (1) is nearly identical.