>>11459715if you are accepting the existence of an additive identity 0 and the field axioms, then you must accept 1=1 since in a field, for any x in F, 1*x=x. And so it holds for x=1 since 1 must be in F for it to be a field. suppose now that there was another y in F such that y*x=x. then using the facts 1*x=x and y*x=x (also the existence of 1/x for x not zero, which is guaranteed by field axioms) you can prove:
1=x*(1/x)=(y*x)(1/x)=y*(x*1/x)=y*1=y and so 1 is unique. this way of proving uniqueness of 1 is far superior than your incoherent mess and if you want to see similar neat proofs or learn this method of thinking, pick up baby rudin
