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Step * 1 of Lemma compatible-half-cubes

.....assertion..... 
1. k : ℕ
2. a : ℚCube(k)
3. b : ℚCube(k)
4. c : ℚCube(k)
5. d : ℚCube(k)
6. ∀i:ℕk. (↑is-half-interval(c i;a i))
7. ∀i:ℕk. (↑is-half-interval(d i;b i))
8. (↑Inhabited(a ⋂ b)) ⇒ (a ⋂ b ≤ a ∧ a ⋂ b ≤ b)
9. ∀i:ℕk. (↑Inhabited(c i ⋂ d i))
⊢ ↑Inhabited(a ⋂ b)
BY
{ (InstLemma `inhabited-intersection-half-cubes` [⌜k⌝;⌜a⌝;⌜b⌝;⌜c⌝;⌜d⌝]⋅
   THEN Auto
   THEN Try ((RWO "assert-is-half-cube" 0 THEN Auto))
   THEN (RWO "assert-inhabited-rat-cube" 0 THENA Auto)
   THEN RepUR ``rat-cube-intersection`` 0
   THEN Auto) }

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Latex:

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Latex:
.....assertion..... 
1.  k  :  \mBbbN{}
2.  a  :  \mBbbQ{}Cube(k)
3.  b  :  \mBbbQ{}Cube(k)
4.  c  :  \mBbbQ{}Cube(k)
5.  d  :  \mBbbQ{}Cube(k)
6.  \mforall{}i:\mBbbN{}k.  (\muparrow{}is-half-interval(c  i;a  i))
7.  \mforall{}i:\mBbbN{}k.  (\muparrow{}is-half-interval(d  i;b  i))
8.  (\muparrow{}Inhabited(a  \mcap{}  b))  {}\mRightarrow{}  (a  \mcap{}  b  \mleq{}  a  \mwedge{}  a  \mcap{}  b  \mleq{}  b)
9.  \mforall{}i:\mBbbN{}k.  (\muparrow{}Inhabited(c  i  \mcap{}  d  i))
\mvdash{}  \muparrow{}Inhabited(a  \mcap{}  b)


By

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Latex:
(InstLemma  `inhabited-intersection-half-cubes`  [\mkleeneopen{}k\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{};\mkleeneopen{}d\mkleeneclose{}]\mcdot{}
  THEN  Auto
  THEN  Try  ((RWO  "assert-is-half-cube"  0  THEN  Auto))
  THEN  (RWO  "assert-inhabited-rat-cube"  0  THENA  Auto)
  THEN  RepUR  ``rat-cube-intersection``  0
  THEN  Auto)

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