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Step * 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. r0≤p @ [right-endpoint(I)][0] - left-endpoint(I)≤partition-mesh(I;p)
8. r0≤[left-endpoint(I) / (p @ [right-endpoint(I)])][||p|| + 1] - [left-endpoint(I) / 
                                                                   (p @ [right-endpoint(I)])][||p||]≤partition-mesh(I;p)
⊢ ((¬0 < ||p||) ⇒ r0≤right-endpoint(I) - left-endpoint(I)≤partition-mesh(I;p))
∧ (0 < ||p||
  ⇒ (r0≤p[0] - left-endpoint(I)≤partition-mesh(I;p)
     ∧ (∀i:ℕ||p|| - 1. r0≤p[i + 1] - p[i]≤partition-mesh(I;p))
     ∧ r0≤right-endpoint(I) - last(p)≤partition-mesh(I;p)))
BY
{ D 0 }

1
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. r0≤p @ [right-endpoint(I)][0] - left-endpoint(I)≤partition-mesh(I;p)
8. r0≤[left-endpoint(I) / (p @ [right-endpoint(I)])][||p|| + 1] - [left-endpoint(I) / 
                                                                   (p @ [right-endpoint(I)])][||p||]≤partition-mesh(I;p)
⊢ (¬0 < ||p||) ⇒ r0≤right-endpoint(I) - left-endpoint(I)≤partition-mesh(I;p)

2
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. r0≤p @ [right-endpoint(I)][0] - left-endpoint(I)≤partition-mesh(I;p)
8. r0≤[left-endpoint(I) / (p @ [right-endpoint(I)])][||p|| + 1] - [left-endpoint(I) / 
                                                                   (p @ [right-endpoint(I)])][||p||]≤partition-mesh(I;p)
⊢ 0 < ||p||
⇒ (r0≤p[0] - left-endpoint(I)≤partition-mesh(I;p)
   ∧ (∀i:ℕ||p|| - 1. r0≤p[i + 1] - p[i]≤partition-mesh(I;p))
   ∧ r0≤right-endpoint(I) - last(p)≤partition-mesh(I;p))

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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.  r0\mleq{}p  @  [right-endpoint(I)][0]  -  left-endpoint(I)\mleq{}partition-mesh(I;p)
8.  r0\mleq{}[left-endpoint(I)  /  (p  @  [right-endpoint(I)])][||p||  +  1] 
-  [left-endpoint(I)  /  (p  @  [right-endpoint(I)])][||p||]\mleq{}partition-mesh(I;p)
\mvdash{}  ((\mneg{}0  <  ||p||)  {}\mRightarrow{}  r0\mleq{}right-endpoint(I)  -  left-endpoint(I)\mleq{}partition-mesh(I;p))
\mwedge{}  (0  <  ||p||
    {}\mRightarrow{}  (r0\mleq{}p[0]  -  left-endpoint(I)\mleq{}partition-mesh(I;p)
          \mwedge{}  (\mforall{}i:\mBbbN{}||p||  -  1.  r0\mleq{}p[i  +  1]  -  p[i]\mleq{}partition-mesh(I;p))
          \mwedge{}  r0\mleq{}right-endpoint(I)  -  last(p)\mleq{}partition-mesh(I;p)))


By

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Latex:
D  0

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