# Left-extensibility-stable implies intermediate subgroup condition

This article gives the statement and possibly, proof, of an implication relation between two subgroup metaproperties. That is, it states that every subgroup satisfying the first subgroup metaproperty (i.e., Left-extensibility-stable subgroup property (?)) must also satisfy the second subgroup metaproperty (i.e., Intermediate subgroup condition (?))

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## Statement

Let be a left-extensibility-stable subgroup property, viz a subgroup property that can be written as where is an extensibility-stable function property.

Then, satisfies the intermediate subgroup condition, or equivalently, whenever with satisfying in the whole of , also satisfies in .

## Examples

Some examples of subgroup properties that are left-extensibility-stable and, on account of this, satisfy the intermediate subgroup condition, are:

## Proof

Let be a left-extensibility-stable subgroup property and be groups such that satisfies in . We need to show that satisfies in .

Let be a function on satisfying property in . Then, we need to show that restricts to a function on which satisfies in .

Since is an extensibility-stable function property, there exists a function on whose restriction to is . Now, since satisfies property in , the restriction of to is well-defined and satisfies property in .

But the restriction to of is the same as the restriction to of . Hence, we have shown that the restriction to of is well-defined and satisfies property over .