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Step * 1 of Lemma face-forall-implies-1

.....subterm..... T:t
1:n
1. H : CubicalSet{j}
2. phi : {H.𝕀 ⊢ _:𝔽}
3. X : Top
4. I : fset(ℕ)
5. rho : H(I)
6. (∀ phi)(rho) = 1 ∈ Point(face_lattice(I))
7. (new-name(I)1) ∈ I ⟶ I+new-name(I)
8. <new-name(I)> ∈ 𝕀(s(rho))
9. (phi(<s(rho), <new-name(I)>>))<(new-name(I)1)>
= phi(<(new-name(I)1)(s(rho)), (<new-name(I)> s(rho) (new-name(I)1))>)
∈ Point(face_lattice(I))
⊢ rho = (new-name(I)1)(s(rho)) ∈ H(I)
BY
{ ((GenConclTerm ⌜new-name(I)⌝⋅ THENA Auto) THEN D -2 THEN Auto THEN RWO  "s-comp-nc-1" 0 THEN Auto) }

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Latex:

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Latex:
.....subterm.....  T:t
1:n
1.  H  :  CubicalSet\{j\}
2.  phi  :  \{H.\mBbbI{}  \mvdash{}  \_:\mBbbF{}\}
3.  X  :  Top
4.  I  :  fset(\mBbbN{})
5.  rho  :  H(I)
6.  (\mforall{}  phi)(rho)  =  1
7.  (new-name(I)1)  \mmember{}  I  {}\mrightarrow{}  I+new-name(I)
8.  <new-name(I)>  \mmember{}  \mBbbI{}(s(rho))
9.  (phi(<s(rho),  <new-name(I)>>))<(new-name(I)1)>
=  phi(<(new-name(I)1)(s(rho)),  (<new-name(I)>  s(rho)  (new-name(I)1))>)
\mvdash{}  rho  =  (new-name(I)1)(s(rho))


By

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Latex:
((GenConclTerm  \mkleeneopen{}new-name(I)\mkleeneclose{}\mcdot{}  THENA  Auto)  THEN  D  -2  THEN  Auto  THEN  RWO    "s-comp-nc-1"  0  THEN  Auto)

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