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Step * 1 1 2 1 2 1 1 1 of Lemma adjacent-partition-points


1. I : Interval
2. icompact(I)
3. p : partition(I)
4. ∀[i:ℕ||full-partition(I;p)|| - 1]
     (frs-non-dec(full-partition(I;p)) ⇒ r0≤full-partition(I;p)[i + 1] - full-partition(I;p)[i]≤partition-mesh(I;p))
5. frs-non-dec(full-partition(I;p))
6. ∀i:ℕ||p|| + 1. r0≤full-partition(I;p)[i + 1] - full-partition(I;p)[i]≤partition-mesh(I;p)
7. 0 < ||p||
8. i : ℕ||p|| - 1
9. r0≤full-partition(I;p)[(i + 1) + 1] - full-partition(I;p)[i + 1]≤partition-mesh(I;p)
⊢ r0≤p[i + 1] - p[i]≤partition-mesh(I;p)
BY
{ (NthHypEq (-1)
   THEN RepeatFor 2 ((EqCD THEN Auto))
   THEN RepUR ``full-partition`` 0
   THEN (RWO "select_cons_tl" 0 THEN Auto' THEN (RWO "select_append_front" 0 THEN Auto')⋅)⋅) }

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Latex:

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Latex:

1.  I  :  Interval
2.  icompact(I)
3.  p  :  partition(I)
4.  \mforall{}[i:\mBbbN{}||full-partition(I;p)||  -  1]
          (frs-non-dec(full-partition(I;p))
          {}\mRightarrow{}  r0\mleq{}full-partition(I;p)[i  +  1]  -  full-partition(I;p)[i]\mleq{}partition-mesh(I;p))
5.  frs-non-dec(full-partition(I;p))
6.  \mforall{}i:\mBbbN{}||p||  +  1.  r0\mleq{}full-partition(I;p)[i  +  1]  -  full-partition(I;p)[i]\mleq{}partition-mesh(I;p)
7.  0  <  ||p||
8.  i  :  \mBbbN{}||p||  -  1
9.  r0\mleq{}full-partition(I;p)[(i  +  1)  +  1]  -  full-partition(I;p)[i  +  1]\mleq{}partition-mesh(I;p)
\mvdash{}  r0\mleq{}p[i  +  1]  -  p[i]\mleq{}partition-mesh(I;p)


By

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Latex:
(NthHypEq  (-1)
  THEN  RepeatFor  2  ((EqCD  THEN  Auto))
  THEN  RepUR  ``full-partition``  0
  THEN  (RWO  "select\_cons\_tl"  0  THEN  Auto'  THEN  (RWO  "select\_append\_front"  0  THEN  Auto')\mcdot{})\mcdot{})

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