abc=1 does not imply a,b,c is -1 or +1 as these are real numbers, e.g. a=1/3, b=2, c=3/2 , abc=1
As for the other question, a,b,c>0 is not an assumption. It follows from the given.
Hmm, yea, you are right. But can you explain a bit how you got a,b,c>0 from given?