Can some tell me if this proof works and if not correct me please.
a+b = a+c -> b=c
~a·a+b = ~a·a+c -> b=c
b=c -> b=c
Proved!
Your proof is not correct. You need to keep in mind that we are dealing in additive notation, so the inverse must be expressed in additive notation (i.e. $\displaystyle a^{-1}\implies -a$)
It should be $\displaystyle a+b=a+c\implies (-a)+a+b=(-a)+a+c\implies(-a+a)+b=(-a+a)+c\implies b=c$
Does this make sense?