Elementary number theory problem about finding a square number with digits with certain properties

Everything is fine upto

(m+1)(m-1)=1001x

After that, you asume that it implies $$(m+1=1001 \land m-1 = x) \lor (m+1=x \land m-1=1001)$$

This is not true, for instance:

11*9=99=33*3, yet, we don't have 33=11 nor 9=3