### Nuprl Lemma : rabs-difference-bound-iff

`∀x,y,z:ℝ.  (|x - y| < z `⇐⇒` ((y - z) < x) ∧ (x < (y + z)))`

Proof

Definitions occuring in Statement :  rless: `x < y` rabs: `|x|` rsub: `x - y` radd: `a + b` real: `ℝ` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` top: `Top` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` rev_implies: `P `` Q` uimplies: `b supposing a` itermConstant: `"const"` req_int_terms: `t1 ≡ t2` false: `False` not: `¬A` uiff: `uiff(P;Q)` prop: `ℙ` cand: `A c∧ B`
Lemmas referenced :  rabs-as-rmax rmax_strict_lb rsub_wf rminus_wf rless-implies-rless real_term_polynomial itermSubtract_wf itermVar_wf itermMinus_wf int-to-real_wf real_term_value_const_lemma real_term_value_sub_lemma real_term_value_var_lemma real_term_value_minus_lemma req-iff-rsub-is-0 radd_wf itermAdd_wf real_term_value_add_lemma rless_wf rmax_wf real_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity cut introduction extract_by_obid sqequalHypSubstitution sqequalTransitivity computationStep isectElimination thin isect_memberEquality voidElimination voidEquality hypothesis lambdaFormation independent_pairFormation dependent_functionElimination hypothesisEquality productElimination independent_functionElimination independent_isectElimination natural_numberEquality computeAll lambdaEquality int_eqEquality intEquality because_Cache productEquality

Latex:
\mforall{}x,y,z:\mBbbR{}.    (|x  -  y|  <  z  \mLeftarrow{}{}\mRightarrow{}  ((y  -  z)  <  x)  \mwedge{}  (x  <  (y  +  z)))

Date html generated: 2017_10_03-AM-08_39_15
Last ObjectModification: 2017_07_28-AM-07_30_41

Theory : reals

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