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Step * 1 1 1 1 of Lemma member-rat-complex-subdiv


1. k : ℕ
2. K : {c:ℚCube(k)| ↑Inhabited(c)}  List
3. c : ℚCube(k)
4. l : ℚCube(k) List
5. y : {c:ℚCube(k)| ↑Inhabited(c)} 
6. (y ∈ K)
7. l = half-cubes-of(k;y) ∈ (ℚCube(k) List)
8. i : ℕ
9. i < ||half-cubes-of(k;y)||
10. c = half-cubes-of(k;y)[i] ∈ ℚCube(k)
11. (y ∈ K)
⊢ ↑is-half-cube(k;c;y)
BY
{ ((RWO "-2" 0 THEN Auto) THEN (GenConclTerm ⌜half-cubes-of(k;y)⌝⋅ THENA Auto)) }

1
1. k : ℕ
2. K : {c:ℚCube(k)| ↑Inhabited(c)}  List
3. c : ℚCube(k)
4. l : ℚCube(k) List
5. y : {c:ℚCube(k)| ↑Inhabited(c)} 
6. (y ∈ K)
7. l = half-cubes-of(k;y) ∈ (ℚCube(k) List)
8. i : ℕ
9. i < ||half-cubes-of(k;y)||
10. c = half-cubes-of(k;y)[i] ∈ ℚCube(k)
11. (y ∈ K)
12. v : {L:ℚCube(k) List| no_repeats(ℚCube(k);L) ∧ (∀h:ℚCube(k). ((h ∈ L) ⇐⇒ ↑is-half-cube(k;h;y)))} 
13. half-cubes-of(k;y)
= v
∈ {L:ℚCube(k) List| no_repeats(ℚCube(k);L) ∧ (∀h:ℚCube(k). ((h ∈ L) ⇐⇒ ↑is-half-cube(k;h;y)))} 
⊢ ↑is-half-cube(k;v[i];y)

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Latex:

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Latex:

1.  k  :  \mBbbN{}
2.  K  :  \{c:\mBbbQ{}Cube(k)|  \muparrow{}Inhabited(c)\}    List
3.  c  :  \mBbbQ{}Cube(k)
4.  l  :  \mBbbQ{}Cube(k)  List
5.  y  :  \{c:\mBbbQ{}Cube(k)|  \muparrow{}Inhabited(c)\} 
6.  (y  \mmember{}  K)
7.  l  =  half-cubes-of(k;y)
8.  i  :  \mBbbN{}
9.  i  <  ||half-cubes-of(k;y)||
10.  c  =  half-cubes-of(k;y)[i]
11.  (y  \mmember{}  K)
\mvdash{}  \muparrow{}is-half-cube(k;c;y)


By

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Latex:
((RWO  "-2"  0  THEN  Auto)  THEN  (GenConclTerm  \mkleeneopen{}half-cubes-of(k;y)\mkleeneclose{}\mcdot{}  THENA  Auto))

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