### Nuprl Lemma : real-continuity2

`∀a,b:ℝ.`
`  ∀f:[a, b] ⟶ℝ`
`    ((∀x,y:{x:ℝ| x ∈ [a, b]} .  (f x ≠ f y `` x ≠ y))`
`    `` (∀k:ℕ+. ∃d:{d:ℝ| r0 < d} . ∀x,y:{x:ℝ| x ∈ [a, b]} .  ((|x - y| ≤ d) `` (|(f x) - f y| ≤ (r1/r(k)))))) `
`  supposing a ≤ b`

Proof

Definitions occuring in Statement :  rfun: `I ⟶ℝ` rccint: `[l, u]` i-member: `r ∈ I` rdiv: `(x/y)` rneq: `x ≠ y` rleq: `x ≤ y` rless: `x < y` rabs: `|x|` rsub: `x - y` int-to-real: `r(n)` real: `ℝ` nat_plus: `ℕ+` uimplies: `b supposing a` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` apply: `f a` natural_number: `\$n`
Definitions unfolded in proof :  top: `Top` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` squash: `↓T` less_than: `a < b` nat_plus: `ℕ+` sq_exists: `∃x:A [B[x]]` rless: `x < y` or: `P ∨ Q` rneq: `x ≠ y` iff: `P `⇐⇒` Q` rfun: `I ⟶ℝ` real-fun: `real-fun(f;a;b)` prop: `ℙ` real: `ℝ` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` false: `False` implies: `P `` Q` not: `¬A` and: `P ∧ Q` le: `A ≤ B` rnonneg: `rnonneg(x)` rleq: `x ≤ y` uimplies: `b supposing a` member: `t ∈ T` all: `∀x:A. B[x]` real-sfun: `real-sfun(f;a;b)` real-cont: `real-cont(f;a;b)`
Lemmas referenced :  int_formula_prop_wf int_term_value_constant_lemma int_term_value_var_lemma int_term_value_add_lemma int_formula_prop_less_lemma itermConstant_wf itermVar_wf itermAdd_wf intformless_wf full-omega-unsat nat_plus_properties req_weakening rneq_functionality rneq_wf not-rneq req_wf i-member_wf rleq_wf rccint_wf rfun_wf real-sfun_wf nat_plus_wf real_wf rsub_wf less_than'_wf real-continuity-ext
Rules used in proof :  voidEquality isect_memberEquality intEquality int_eqEquality dependent_pairFormation approximateComputation imageElimination unionElimination independent_functionElimination because_Cache setEquality independent_isectElimination equalitySymmetry equalityTransitivity axiomEquality natural_numberEquality minusEquality rename setElimination applyEquality isectElimination voidElimination independent_pairEquality productElimination lambdaEquality isect_memberFormation hypothesisEquality thin dependent_functionElimination sqequalHypSubstitution hypothesis lambdaFormation extract_by_obid introduction cut computationStep sqequalTransitivity sqequalReflexivity sqequalRule sqequalSubstitution

Latex:
\mforall{}a,b:\mBbbR{}.
\mforall{}f:[a,  b]  {}\mrightarrow{}\mBbbR{}
((\mforall{}x,y:\{x:\mBbbR{}|  x  \mmember{}  [a,  b]\}  .    (f  x  \mneq{}  f  y  {}\mRightarrow{}  x  \mneq{}  y))
{}\mRightarrow{}  (\mforall{}k:\mBbbN{}\msupplus{}
\mexists{}d:\{d:\mBbbR{}|  r0  <  d\}
\mforall{}x,y:\{x:\mBbbR{}|  x  \mmember{}  [a,  b]\}  .    ((|x  -  y|  \mleq{}  d)  {}\mRightarrow{}  (|(f  x)  -  f  y|  \mleq{}  (r1/r(k))))))
supposing  a  \mleq{}  b

Date html generated: 2018_05_22-PM-02_11_26
Last ObjectModification: 2018_05_21-AM-00_27_05

Theory : reals

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