# Question 10674: Inclusive Membership in a Declared Closed Rectangle

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0020-10674 · q-20261013-advance-20261023-r07-multivariable-real-function-0049



## Original question and givens

Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-25, 25, -24, 26] and point [x,y]=[-1, 1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.


- G001 — Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-25, 25, -24, 26] and point [x,y]=[-1, 1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-multivariable-real-function](https://ocme.evemisslab.com/en/mko/mko-multivariable-real-function/)





## Step-by-step derivation

S001 · calculationBefore

rectangle coordinate0 lower slack := sub(-1, -25)

After

rectangle coordinate0 lower slack = 24

Conditions: none

Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001





S002 · calculationBefore

rectangle coordinate0 upper slack := sub(25, -1)

After

rectangle coordinate0 upper slack = 26

Conditions: none

Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · impliesBefore

coordinate index=0;value=-1;closed bounds=[-25, 25]

After

signed slacks=[24, 26];inside inclusive interval=Yes

Conditions: none

Equal bounds are valid and endpoints are included.Both coordinates must pass;the domain restriction is not erased by a polynomial's possible global definition.

Step identifiers and dependenciesRule: rectangle_both_coordinate_closed_guards · Uses: G001, S002





S004 · calculationBefore

rectangle coordinate1 lower slack := sub(1, -24)

After

rectangle coordinate1 lower slack = 25

Conditions: none

Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

rectangle coordinate1 upper slack := sub(26, 1)

After

rectangle coordinate1 upper slack = 25

Conditions: none

Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

coordinate index=1;value=1;closed bounds=[-24, 26]

After

signed slacks=[25, 25];inside inclusive interval=Yes

Conditions: none

Equal bounds are valid and endpoints are included.Both coordinates must pass;the domain restriction is not erased by a polynomial's possible global definition.

Step identifiers and dependenciesRule: rectangle_both_coordinate_closed_guards · Uses: G001, S005









## Reverse checks


- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Recompute rectangle coordinate1 upper slack with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute rectangle coordinate1 lower slack with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Recompute rectangle coordinate0 upper slack with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Recompute rectangle coordinate0 lower slack with exact rational arithmetic, not rounded decimals.





## Result

([-25, 25, -24, 26], [-1, 1], Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: d0bd9595031217e45f605ec9ca0972998802f0adfe4f3e86bd7913b647ed41f5
- candidate record SHA-256: d3dc84c14849fe95b5382e95a7b83e2700d264037078b94b25fe3950e326a869
- semantic review receipt SHA-256: b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598f
- question-first witness SHA-256: 3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261023-r07-multivariable-real-function-0049.json)