I must say I didn't see either what you did wrong, the only possible thing I can see is that you started with the thing you where going to prove and developed it into something obvious, instead of doing the opposite, that was maybe what ThePrefectHacker ment by the preservation of the not equality.
My secret is simple. Pure math. What do I mean by that? Occasionally everyone (I mean mathemations) are lazy (like usual) and you do not state out all the details. For example, the proper procedure to use the substitution rule for integrals is to show the inner function is differenciable and then show that the outer function poses an anti-derivative.... the traditional stuff. Most people ignore that (I do that sometimes) and make a mistake. When doing a math problem we need to go all the way back to the definitions. This is why Mathematical proves (like Perelman's) take an awufully long time to check. Because everyone returns back to the defintions of a topology and procede from there. This is why many mathemations did make mistakes (but easily correctable) in their long proofs because they were far too lazy to do that from the beginning.How he manages this I have yet to discover. I suspect a complete dissection of his brain would provide the answer...
is true because 1 x 1 = 1
is true because -1 x -1 = 1
It does not follow that 1 must equal -1 as square root is not a 1 to 1 function
Plus assuming what you want to be true implies something that is true does not make original assumption true or else
All multiplications give the answer zero
6 x 0 = 0
Second line is true and follows from first line. However first line certainly isn't true
Edit: Okey, I looked up the definition of square root and saw it could be both the values. According to wikipedia, there is the principal square root, apperantly, and the other square root.
Another thing I didn't quite understand was if the symbol denoted the principal square root or not. First it says it does, then it says that "The square root symbol ( ) was first used during the 16th century".
You use the sign to show you are considering both possible solutions