# Find a lambda term satisfying two equations

I'm just looking for the general idea on how to approach the following problem:

Find a term $$\Delta=\lambda x.xUV$$ such that:

1. $$\Delta\Delta=K$$
2. $$\Delta K=S$$

(it's a system of 2 equations, I didn't know how to format it properly)

Where $$K=\lambda xy.x$$ and $$S=\lambda xyz.xz(yz)$$

I think I need to use Bohm theorem but I don't know exactly how.

This is not directly a consequence of Böhm's theorem. Böhm's theorem states that for any strongly normalizing $$A$$ and $$B$$ that are not $$\beta\eta$$-convertible, there exists $$\Gamma$$ such that $$\Gamma A = K$$ and $$\Gamma B = S$$. You don't get to constrain $$\Gamma = A$$ or the required form $$\lambda x. x U V$$. There may be a way to use Böhm's theorem on a different term, but this particular system of equations can easily be solved manually.
If such a term exists then $$\Delta K = K U V = U$$, so to satisfy the second equation it is necessary that $$U = S$$.
Then $$\Delta \Delta = (\lambda x. x S V) (\lambda x. x S V) = S S V V = S V (V V) = \lambda y. V y (V V y)$$. To satisfy the first equation, $$V y (V V y) = K y = \lambda z. y$$. One way to satisfy this is to take $$V = \lambda y z z'. y$$.
Finally, verify that $$\Delta = \lambda x. x S (\lambda y z z'. y)$$ satisfies the two equations.