Extending Operation is a Slowly Progressing Mapping

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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.

$\blacksquare$


Sources