### Nuprl Lemma : real-continuity1-ext

`∀a,b:ℝ.`
`  ∀f:[a, b] ⟶ℝ`
`    ∀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 ∀x,y:{x:ℝ| x ∈ [a, b]} .  ((x = y) `` ((f x) = (f y))) `
`  supposing a ≤ b`

Proof

Definitions occuring in Statement :  rfun: `I ⟶ℝ` rccint: `[l, u]` i-member: `r ∈ I` rdiv: `(x/y)` rleq: `x ≤ y` rless: `x < y` rabs: `|x|` rsub: `x - y` req: `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 :  real-continuity-ext real-continuity1 so_apply: `x[s]` so_lambda: `λ2x.t[x]` guard: `{T}` implies: `P `` Q` all: `∀x:A. B[x]` sq_type: `SQType(T)` unit: `Unit` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` ifthenelse: `if b then t else f fi ` bfalse: `ff` it: `⋅` btrue: `tt` subtract: `n - m` member: `t ∈ T`
Lemmas referenced :  product_subtype_base unit_subtype_base unit_wf2 trivial-equal bool_subtype_base bool_wf subtype_base_sq real-continuity1 real-continuity-ext
Rules used in proof :  independent_pairEquality lambdaFormation lambdaEquality productEquality inrEquality because_Cache independent_functionElimination dependent_functionElimination natural_numberEquality axiomEquality inlEquality independent_isectElimination cumulativity isectElimination equalitySymmetry equalityTransitivity sqequalHypSubstitution thin sqequalRule hypothesis extract_by_obid instantiate cut sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution introduction

Latex:
\mforall{}a,b:\mBbbR{}.
\mforall{}f:[a,  b]  {}\mrightarrow{}\mBbbR{}
\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  \mforall{}x,y:\{x:\mBbbR{}|  x  \mmember{}  [a,  b]\}  .    ((x  =  y)  {}\mRightarrow{}  ((f  x)  =  (f  y)))
supposing  a  \mleq{}  b

Date html generated: 2018_05_22-PM-02_12_07
Last ObjectModification: 2018_05_21-AM-00_29_07

Theory : reals

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