# Question 34049: Membership Including Closed Endpoints

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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.

R0067-34049 · q-20261005-advance-20261006-r01-order-relation-0074



## Original question and givens

Under the usual real order, is (101)/4 in the closed interval [-23, 25]? The denominator is positive and both endpoints are included.


- G001 — Under the usual real order, is (101)/4 in the closed interval [-23, 25]? The denominator is positive and both endpoints are included.





## 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-order-relation](https://ocme.evemisslab.com/en/mko/mko-order-relation/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-order-closed-interval, [lower, operation, p, q, upper]]

After

derived = [closed_interval, no-invented-domain]

Conditions: none

Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations.

Step identifiers and dependenciesRule: R67_SOURCE_specific_keys_operation_domain_and_numeric_bank · Uses: G001





S002 · calculationBefore

Interval ORIGINAL lower times positive denominator := mul(-23, 4)

After

Interval ORIGINAL lower times positive denominator = -92

Conditions: none

Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.

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





S003 · calculationBefore

Interval ORIGINAL upper times positive denominator := mul(25, 4)

After

Interval ORIGINAL upper times positive denominator = 100

Conditions: none

Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.

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





S004 · impliesBefore

source roles = [101, 4, -23, 25]

After

derived = [-92, 101, 100, Yes, No, No]

Conditions: none

For a positive denominator,lower≤p/q≤upper is equivalent to lower*q≤p≤upper*q.BOTH endpoints are included.A singleton is valid but contains only its endpoint.Do not use open inequalities or silently sort reversed endpoints.

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









## Reverse checks


- S004 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
- S003 · premise_and_result_audit — Recompute Interval ORIGINAL upper times positive denominator with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute Interval ORIGINAL lower times positive denominator with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.





## Result

No

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: b10e2ba5e619c298887a764df572804b691ed1c3c0f165ae083435faf13b0480
- candidate record SHA-256: 3aa80761e04a5333ed29fabc36e55c2013d486b39634824a4b7bb9bea92d0772
- semantic review receipt SHA-256: c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c
- question-first witness SHA-256: 12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261005-advance-20261006-r01-order-relation-0074.json)