# Question 34051: Order after a Common Addend [Return to catalogue page 171](https://ocme.evemisslab.com/en/questions/page/171/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261005-advance-20261006-r01-order-relation-foundations/#q-20261005-advance-20261006-r01-order-relation-0076) 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. R0067-34051 · q-20261005-advance-20261006-r01-order-relation-0076 ## Original question and givens Under the usual real order, a=0, b=0, and c=-12. Compare a+c and b+c, giving <, =, or > in that order. - G001 — Under the usual real order, a=0, b=0, and c=-12. Compare a+c and b+c, giving <, =, or > in that order. ## 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-common-translation, [a, b, c, operation]] After derived = [common_translation, 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 Translation ORIGINAL a plus same c := sum(0, -12) After Translation ORIGINAL a plus same c = -12 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 Translation ORIGINAL b plus same c := sum(0, -12) After Translation ORIGINAL b plus same c = -12 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 · calculationBefore Translation ORIGINAL translated left minus right := sub(-12, -12) After Translation ORIGINAL translated left minus right = 0 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, S003 S005 · impliesBefore source roles = [0, 0, -12] After derived = [-12, -12, 0, =] Conditions: none Adding the SAME c to BOTH sides preserves difference a−b.Subtract c from both to recover the original comparison,including equality and negative c.Never translate only one side;derived sums may exceed SAFE. Step identifiers and dependenciesRule: Order_COMMON_addend_cancel_BOTH_sides · Uses: G001, S004 ## Reverse checks - S005 · 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. - S004 · premise_and_result_audit — Recompute Translation ORIGINAL translated left minus right with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Translation ORIGINAL b plus same c with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Translation ORIGINAL a plus same c 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 = 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: e78673acee229d18b5edbae97e65e28cfb1f83d9d381679e58ab88976d1c351c - candidate record SHA-256: c67eb51cacc703be108d83561b13e5caf802a220e800263ed7b85e55fcd1f036 - 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-0076.json)