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added within_continuous_patch #1883

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6be0f8a
added within_continuous_patch
holgerthies Mar 5, 2026
e0e1e4e
changelog
holgerthies Mar 5, 2026
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3 changes: 3 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Original file line number Diff line number Diff line change
Expand Up @@ -70,6 +70,9 @@
- in `derive.v`:
+ lemmas `compact_EVT_max`, `compact_EVT_min`, `EVT_max_rV`, `EVT_min_rV`

- in `subspace_topology.v`:
+ lemma `within_continuous_patch`

### Changed

- in `constructive_ereal.v`: fixed the infamous `%E` scope bug.
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24 changes: 24 additions & 0 deletions theories/topology_theory/subspace_topology.v
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Expand Up @@ -705,3 +705,27 @@ HB.instance Definition _ :=
@isContinuous.Build (subspace A) Z (g \o f) continuous_comp_subproof.

End continuous_fun_comp.

Section continuous_patch.
Context {U : topologicalType} {V : topologicalType}.
Variables (A : set U) (B : set U) (f : U -> V) (g : U -> V).
Hypothesis cont1 : {within A, continuous f}.
Hypothesis cont2 : {within B, continuous g}.
Hypothesis closedA : closed A.
Hypothesis closedB : closed B.
Hypothesis eq_AB : forall x, x \in A `&` B -> f x = g x.

Lemma within_continuous_patch : {within A `|` B, continuous (patch g A f)}.
Proof.
apply: withinU_continuous => //.
have : {in A, f =1 patch g A f } by rewrite /patch => r ->.
by move/subspace_eq_continuous; apply.
have : {in B, g =1 patch g A f }.
move=> r rab.
rewrite /patch; case: ifPn => [xab | //].
apply/esym/eq_AB.
by rewrite inE; split; apply set_mem.
by move/subspace_eq_continuous; apply.
Qed.

End continuous_patch.
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