Subspace of Product Space is Homeomorphic to Factor Space/Proof 2

Proof
Let $z \in X$.

Let $i \in I$.

Let $Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$.

Let $\upsilon_i$ be the subspace topology of $Y_i$ relative to $\tau$.

Let $p_i = \pr_i {\restriction_{Y_i}}$.

Note that by definitions of a restriction and a projection then:
 * $\forall y \in Y_i: \map {p_i} y = y_i$