# How to deduce the following map between Zariski tangent spaces is surjective?

Let $$f: X \rightarrow Y$$ be a morphism of schemes: $$X$$, $$Y$$ are regular schemes, let $$Z_1, Z_2$$ be two closed regular subschemes of $$X$$, let $$x \in X \times_Y Z_2$$ such that $$y = f(x) \in Z_1 \cap Z_2$$. Then I would like to know how I can deduce that the map $$T_xX \rightarrow (T_yY/T_yZ_2) \otimes_{k(y)} k(x)$$ is surjective knowing that

1) the image of $$T_{x} (X \times_Y Z_1)$$ generates $$T_yY/T_yZ_2$$

2) $$T_x(X \times_Y Z_1) \rightarrow T_y Z_1 \otimes_{k(y)} k(x)$$ is surjective.

It is supposed to follow from these things but not seeing how... I would appreciate any explanation on this. Thank you!

## This question has an open bounty worth +50 reputation from Johnny T. ending in 7 days.

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• $f(x)\in Y$, but $Z_1\cap Z_2\subset X$, so did you mean something else? – Mohan 5 hours ago