Extending Operation is a Slowly Progressing Mapping

Theorem
Let $S$ denote the class of all ordinal sequences.

Let $E: S \to S$ be an extending operation on $S$.

Then $E$ is a slowly progressing mapping.

Proof
Let $\theta \in S$ be an $\alpha$-sequence.

By definition of extending operation:
 * $\map E \theta = \theta \cup \tuple {\alpha, x}$

where $x$ is arbitrary.

Thus:
 * $\theta \subseteq \map E \theta$

and it is seen that $E$ is by definition a progressing mapping.

It is also seen that:
 * $\card \theta = \card \alpha$

while:
 * $\card {\map E \theta} = \card \alpha + 1$

demonstrating that $E$ is a slowly progressing mapping.