> > Nonsense. Anyone can manipulate algebra inconsistently.
>
> Only if they don't know the rules or choose to ignore them
>
> > a = b, given
> > a^2 = ab, multiply by a
> > a^2- b^2  = ab - b^2, subtract b^2
> > (a+b)(a-b) = b(a-b), factorize
> > a+b = b, cancel a-b.
>
> ... is dividing by (a-b) which, given that a=b is identically equal to
> zero, is an illegitimate operation.
>
> So, the consequent inference
>
> > b+b = b, because a = b, given
> > 2 = 1, divide by b
>
> is, naturally, itself nonsense.

Yes, but was the error in a = b, or the division by zero?
I can easily make use of (a <> b) and legitimately conclude either
a  = 0 or b = 0.