Wolfram Alpha proves that N = N - 1

Not only is one of your premises (a/a = b/b -> a=b) clearly false, but your conclusion doesn't mean anything.

a=b ^ a=b -> n=(n-1) ^ n!=(n-1) is just a contradiction. Your first and second terms together imply n=(n-1) which directly contradicts your third term.

I'm afraid you've simply proved a contradiction, which just means (because of, well, proof by contradiction) that you're arguments are flawed.