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Step * of Lemma rat-cube-third-exists

∀k:ℕ. ∀c:ℚCube(k).  ((↑Inhabited(c)) ⇒ (∃p:ℝ^k. (in-rat-cube(k;p;c) ∧ rat-cube-third(k;p;c))))
BY
{ (Auto
   THEN (RWO "assert-inhabited-rat-cube" (-1) THENA Auto)
   THEN D 0 With ⌜λi.((r(2) * rat2real(fst((c i)))) + rat2real(snd((c i)))/r(3))⌝ 
   THEN Auto
   THEN ((D 0 THENA Auto) THEN Reduce 0)
   THEN (D 3 With ⌜i⌝  THENA Auto)
   THEN MoveToConcl (-1)
   THEN (GenConclTerm ⌜c i⌝⋅ THENA Auto)
   THEN D -2
   THEN RepUR ``inhabited-rat-interval rat-interval-third`` 0
   THEN RW assert_pushdownC 0
   THEN Auto
   THEN ((RWO  "rleq-rat2real<" 7 THENM (DupHyp 7 THEN nRMul ⌜r(2)⌝ (-1)⋅)) THENA Auto)
   THEN nRMul ⌜r(3)⌝ 0⋅
   THEN Auto) }

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Latex:

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Latex:
\mforall{}k:\mBbbN{}.  \mforall{}c:\mBbbQ{}Cube(k).    ((\muparrow{}Inhabited(c))  {}\mRightarrow{}  (\mexists{}p:\mBbbR{}\^{}k.  (in-rat-cube(k;p;c)  \mwedge{}  rat-cube-third(k;p;c))))


By

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Latex:
(Auto
  THEN  (RWO  "assert-inhabited-rat-cube"  (-1)  THENA  Auto)
  THEN  D  0  With  \mkleeneopen{}\mlambda{}i.((r(2)  *  rat2real(fst((c  i))))  +  rat2real(snd((c  i)))/r(3))\mkleeneclose{} 
  THEN  Auto
  THEN  ((D  0  THENA  Auto)  THEN  Reduce  0)
  THEN  (D  3  With  \mkleeneopen{}i\mkleeneclose{}    THENA  Auto)
  THEN  MoveToConcl  (-1)
  THEN  (GenConclTerm  \mkleeneopen{}c  i\mkleeneclose{}\mcdot{}  THENA  Auto)
  THEN  D  -2
  THEN  RepUR  ``inhabited-rat-interval  rat-interval-third``  0
  THEN  RW  assert\_pushdownC  0
  THEN  Auto
  THEN  ((RWO    "rleq-rat2real<"  7  THENM  (DupHyp  7  THEN  nRMul  \mkleeneopen{}r(2)\mkleeneclose{}  (-1)\mcdot{}))  THENA  Auto)
  THEN  nRMul  \mkleeneopen{}r(3)\mkleeneclose{}  0\mcdot{}
  THEN  Auto)

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