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Step * 2 of Lemma general-partition-sum-no-mc


1. I : Interval
2. icompact(I)
3. f : {f:I ⟶ℝ| ifun(f;I)} 
4. e : {e:ℝ| r0 < e} 
5. ∃b:ℝ. ((r0 ≤ b) ∧ (∀x:ℝ. ((x ∈ I) ⇒ (|f x| ≤ b))))
⊢ ∃d:{d:ℝ| r0 < d} 
   ∀p,q:{p:partition(I)| partition-mesh(I;p) ≤ d} . ∀x:partition-choice(full-partition(I;p)).
   ∀y:partition-choice(full-partition(I;q)).
     (|S(f;full-partition(I;q)) - S(f;full-partition(I;p))| ≤ e)
BY
{ (ExRepD THEN InstLemma `general-partition-sum-from-bound` [⌜I⌝;⌜f⌝;⌜b⌝]⋅ THEN Auto) }

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Latex:

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Latex:

1.  I  :  Interval
2.  icompact(I)
3.  f  :  \{f:I  {}\mrightarrow{}\mBbbR{}|  ifun(f;I)\} 
4.  e  :  \{e:\mBbbR{}|  r0  <  e\} 
5.  \mexists{}b:\mBbbR{}.  ((r0  \mleq{}  b)  \mwedge{}  (\mforall{}x:\mBbbR{}.  ((x  \mmember{}  I)  {}\mRightarrow{}  (|f  x|  \mleq{}  b))))
\mvdash{}  \mexists{}d:\{d:\mBbbR{}|  r0  <  d\} 
      \mforall{}p,q:\{p:partition(I)|  partition-mesh(I;p)  \mleq{}  d\}  .  \mforall{}x:partition-choice(full-partition(I;p)).
      \mforall{}y:partition-choice(full-partition(I;q)).
          (|S(f;full-partition(I;q))  -  S(f;full-partition(I;p))|  \mleq{}  e)


By

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Latex:
(ExRepD  THEN  InstLemma  `general-partition-sum-from-bound`  [\mkleeneopen{}I\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{}]\mcdot{}  THEN  Auto)

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