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Step * 1 2 2 of Lemma nearby-partition-choice


1. I : Interval@i
2. icompact(I)
3. p : partition(I)@i
4. q : partition(I)@i
5. e : {e:ℝ| r0 < e} @i
6. nearby-partitions(e;p;q)
7. x : partition-choice(full-partition(I;p))@i
8. nearby-partitions(e;full-partition(I;p);full-partition(I;q))
⊢ ∃y:partition-choice(full-partition(I;q)). ∀i:ℕ||full-partition(I;p)|| - 1. (|x[i] - y[i]| ≤ e)
BY
{ Assert ⌜frs-non-dec(full-partition(I;p)) ∧ frs-non-dec(full-partition(I;q))⌝⋅ }

1
.....assertion..... 
1. I : Interval@i
2. icompact(I)
3. p : partition(I)@i
4. q : partition(I)@i
5. e : {e:ℝ| r0 < e} @i
6. nearby-partitions(e;p;q)
7. x : partition-choice(full-partition(I;p))@i
8. nearby-partitions(e;full-partition(I;p);full-partition(I;q))
⊢ frs-non-dec(full-partition(I;p)) ∧ frs-non-dec(full-partition(I;q))

2
1. I : Interval@i
2. icompact(I)
3. p : partition(I)@i
4. q : partition(I)@i
5. e : {e:ℝ| r0 < e} @i
6. nearby-partitions(e;p;q)
7. x : partition-choice(full-partition(I;p))@i
8. nearby-partitions(e;full-partition(I;p);full-partition(I;q))
9. frs-non-dec(full-partition(I;p)) ∧ frs-non-dec(full-partition(I;q))
⊢ ∃y:partition-choice(full-partition(I;q)). ∀i:ℕ||full-partition(I;p)|| - 1. (|x[i] - y[i]| ≤ e)

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Latex:

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Latex:

1.  I  :  Interval@i
2.  icompact(I)
3.  p  :  partition(I)@i
4.  q  :  partition(I)@i
5.  e  :  \{e:\mBbbR{}|  r0  <  e\}  @i
6.  nearby-partitions(e;p;q)
7.  x  :  partition-choice(full-partition(I;p))@i
8.  nearby-partitions(e;full-partition(I;p);full-partition(I;q))
\mvdash{}  \mexists{}y:partition-choice(full-partition(I;q)).  \mforall{}i:\mBbbN{}||full-partition(I;p)||  -  1.  (|x[i]  -  y[i]|  \mleq{}  e)


By

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Latex:
Assert  \mkleeneopen{}frs-non-dec(full-partition(I;p))  \mwedge{}  frs-non-dec(full-partition(I;q))\mkleeneclose{}\mcdot{}

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