Does removing the (1) in $\Phi{(1)}$ affect the proof that $ K_0 \leq_m K$ or not?

Question

The fragment below from Martin Davis' book shows $ K_0 \leq_m K$ and also proves $ K_0 \leq_1 K $.

My question is if we remove the $(1)$ of $ \Phi^{(1)}$ in the definition of $Y$ (i.e fifth line in picture) does this affect the proof or not? (i.e if we use $Y= \phi (l(X_2),r(X_2))$ is True for $ K_0 \leq_1 K $ proof?) ?