In Exercise 1.9 (iii) in A primer of infinitesimal analysis by John Bell we are asked to show that the following assertion is false: "$x^2+y^2=0$ implies $x^2=0$ for every $x,y\in R$", where $R$ is the fundamental object in any smooth world $\mathbb{S}$, i. e., the smooth real line. Well, for this exercise there is a hint: to use the Principle of Microcancellation, but I just can't even see how to use it to prove that that sentence is false.
Any suggestion, another hint to prove that that sentence is false?
Assume, for the sake of deducing a contradiction, that for any numbers $x$ and $y$ the implication $x^2 + y^2 = 0 \Rightarrow x^2 = 0$ holds.
Then $\varepsilon \cdot \eta = 0$ for all $\varepsilon,\eta \in \Delta$, by the argument in the following paragraph. However, we know that this statement is wrong (for instance from part (i) of the quoted exercise).
So let $\varepsilon,\eta \in \Delta$ be given. Then $(\varepsilon+\eta)^2 + (\varepsilon-\eta)^2 = 2\varepsilon\eta - 2\varepsilon\eta = 0$. Hence by assumption $(\varepsilon+\eta)^2=2\varepsilon\delta = 0$. Assuming that $2$ in cancellable, we obtain $\varepsilon\delta = 0$.