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Step * 1 1 1 of Lemma nearby-separated-partitions


1. I : Interval@i
2. icompact(I)
3. iproper(I)
4. p : partition(I)@i
5. q : partition(I)@i
6. e : {e:ℝ| r0 < e} @i
7. q1 : partition(I)
8. frs-increasing(q1)
9. nearby-partitions(e;p;q1)
10. (e/r(2)) ∈ {e:ℝ| r0 < e} 
11. r : partition(I)
12. frs-increasing(r)
13. nearby-partitions((e/r(2));q;r)
⊢ ∃q':partition(I). (separated-partitions(q1;q') ∧ nearby-partitions(e;p;q1) ∧ nearby-partitions(e;q;q'))
BY
{ (InstLemma `nearby-increasing-partition-avoids` [⌜I⌝;⌜r⌝;⌜q1⌝;⌜(e/r(2))⌝]⋅ THENA Auto) }

1
1. I : Interval@i
2. icompact(I)
3. iproper(I)
4. p : partition(I)@i
5. q : partition(I)@i
6. e : {e:ℝ| r0 < e} @i
7. q1 : partition(I)
8. frs-increasing(q1)
9. nearby-partitions(e;p;q1)
10. (e/r(2)) ∈ {e:ℝ| r0 < e} 
11. r : partition(I)
12. frs-increasing(r)
13. nearby-partitions((e/r(2));q;r)
14. ∃q:partition(I). (frs-increasing(q) ∧ nearby-partitions((e/r(2));r;q) ∧ frs-separated(q;q1))
⊢ ∃q':partition(I). (separated-partitions(q1;q') ∧ nearby-partitions(e;p;q1) ∧ nearby-partitions(e;q;q'))

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Latex:

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Latex:

1.  I  :  Interval@i
2.  icompact(I)
3.  iproper(I)
4.  p  :  partition(I)@i
5.  q  :  partition(I)@i
6.  e  :  \{e:\mBbbR{}|  r0  <  e\}  @i
7.  q1  :  partition(I)
8.  frs-increasing(q1)
9.  nearby-partitions(e;p;q1)
10.  (e/r(2))  \mmember{}  \{e:\mBbbR{}|  r0  <  e\} 
11.  r  :  partition(I)
12.  frs-increasing(r)
13.  nearby-partitions((e/r(2));q;r)
\mvdash{}  \mexists{}q':partition(I)
      (separated-partitions(q1;q')  \mwedge{}  nearby-partitions(e;p;q1)  \mwedge{}  nearby-partitions(e;q;q'))


By

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Latex:
(InstLemma  `nearby-increasing-partition-avoids`  [\mkleeneopen{}I\mkleeneclose{};\mkleeneopen{}r\mkleeneclose{};\mkleeneopen{}q1\mkleeneclose{};\mkleeneopen{}(e/r(2))\mkleeneclose{}]\mcdot{}  THENA  Auto)

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