Complete Factorizations of Proper Element in Principal Ideal Domain are Equivalent

Theorem
Let $\struct {D, +, \circ}$ be a principal ideal domain.

Let $x \in D$ be a proper element of $D$.

Let there be two tidy factorizations of $x$:
 * $x = u_y \circ y_1 \circ y_2 \circ \cdots \circ y_m = F_1$
 * $x = u_z \circ z_1 \circ z_2 \circ \cdots \circ z_n = F_2$

Then $F_1$ and $F_2$ are equivalent.