>Step 1--> prove a^p + b^p != c^p with a < p ,a,b,c, naturals >Step 2--> extend to rationals , still a < p
Step 2 fails.
You can scale an integer non-solution down to get a rational non-solution, but that doesn't prove that there are no rational solutions.
To prove that there are no rational solutions, it's not acceptable logic to start with an assumed integer solution and scale down to a rational one. Rather, you must start by assuming a rational solution and try for a contradiction. Scaling up fails since when scaling rational a with a < p up to integer A, there is no guarantee that A < p, hence no contradiction.
But this has already been explained to you.
Bottom line -- your proof is hopelessly flawed.
Moreover, your logical skills are so weak that there's no possibility that you can prove _anything_ non-tivial relating to _any_ math problem.
Stop wasting your time with mathematical proofs -- your brain isn't wired for that.