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Step * 2 2 2 of Lemma cubical-type-ap-rename-one-equal

.....subterm..... T:t
3:n
1. X : CubicalSet
2. A : {X ⊢ _}
3. I : Cname List
4. x : Cname
5. y : Cname
6. a : X([x / I])
7. u : A(a)
8. v : A(a)
9. ¬(y ∈ I)
10. ¬(x ∈ I)
11. (u a rename-one-name(x;y)) = (v a rename-one-name(x;y)) ∈ A(rename-one-name(x;y)(a))
12. (rename-one-name(x;y) o rename-one-name(y;x)) = 1 ∈ name-morph([x / I];[x / I])
13. ((u a rename-one-name(x;y)) rename-one-name(x;y)(a) rename-one-name(y;x))
= ((v a rename-one-name(x;y)) rename-one-name(x;y)(a) rename-one-name(y;x))
∈ A(a)
⊢ v = ((v a rename-one-name(x;y)) rename-one-name(x;y)(a) rename-one-name(y;x)) ∈ A(a)
BY
{ ((InstLemma `cubical-type-ap-morph-comp` 
    [⌜X⌝;⌜A⌝;⌜[x / I]⌝;⌜[y / I]⌝;⌜[x / I]⌝;⌜rename-one-name(x;y)⌝;⌜rename-one-name(y;x)⌝;⌜a⌝;⌜v⌝]⋅
    THENA Auto
    )
   THEN Symmetry
   THEN NthHypEq (-1)
   THEN EqCD
   THEN Auto) }

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Latex:

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Latex:
.....subterm.....  T:t
3:n
1.  X  :  CubicalSet
2.  A  :  \{X  \mvdash{}  \_\}
3.  I  :  Cname  List
4.  x  :  Cname
5.  y  :  Cname
6.  a  :  X([x  /  I])
7.  u  :  A(a)
8.  v  :  A(a)
9.  \mneg{}(y  \mmember{}  I)
10.  \mneg{}(x  \mmember{}  I)
11.  (u  a  rename-one-name(x;y))  =  (v  a  rename-one-name(x;y))
12.  (rename-one-name(x;y)  o  rename-one-name(y;x))  =  1
13.  ((u  a  rename-one-name(x;y))  rename-one-name(x;y)(a)  rename-one-name(y;x))
=  ((v  a  rename-one-name(x;y))  rename-one-name(x;y)(a)  rename-one-name(y;x))
\mvdash{}  v  =  ((v  a  rename-one-name(x;y))  rename-one-name(x;y)(a)  rename-one-name(y;x))


By

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Latex:
((InstLemma  `cubical-type-ap-morph-comp` 
    [\mkleeneopen{}X\mkleeneclose{};\mkleeneopen{}A\mkleeneclose{};\mkleeneopen{}[x  /  I]\mkleeneclose{};\mkleeneopen{}[y  /  I]\mkleeneclose{};\mkleeneopen{}[x  /  I]\mkleeneclose{};\mkleeneopen{}rename-one-name(x;y)\mkleeneclose{};\mkleeneopen{}rename-one-name(y;x)\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}v\mkleeneclose{}]\mcdot{}
    THENA  Auto
    )
  THEN  Symmetry
  THEN  NthHypEq  (-1)
  THEN  EqCD
  THEN  Auto)

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