Step * 2 1 1 1 1 of Lemma path-eq-equiv

.....assertion..... 
1. CubicalSet
2. {X ⊢ _}
3. {X ⊢ _:A}
4. {X ⊢ _:A}
5. Cname List
6. alpha X(I)
7. z1 Cname
8. ¬(z1 ∈ I)
9. a1 A(iota(z1)(alpha))
10. name-path-endpoints(X;A;a;b;I;alpha;z1;a1)
11. Cname
12. ¬(z ∈ I)
13. b2 named-path(X;A;a;b;I;alpha;z)
14. (a1 iota(z1)(alpha) rename-one-name(z1;z)) b2 ∈ A(iota(z)(alpha))
15. ((a1 iota(z1)(alpha) rename-one-name(z1;z)) rename-one-name(z1;z)(iota(z1)(alpha)) rename-one-name(z;z1))
(a1 iota(z1)(alpha) (rename-one-name(z1;z) rename-one-name(z;z1)))
∈ A((rename-one-name(z1;z) rename-one-name(z;z1))(iota(z1)(alpha)))
16. (rename-one-name(z1;z) rename-one-name(z;z1)) 1 ∈ name-morph([z1 I];[z1 I])
⊢ (iota(z1)(alpha) rename-one-name(z;z1)(iota(z)(alpha)) ∈ X([z1 I]))
∧ (iota(z)(alpha) rename-one-name(z1;z)(iota(z1)(alpha)) ∈ X([z I]))
BY
(D THEN RWO "cube-set-restriction-comp" THEN Auto THEN Symmetry THEN BLemma `rename-one-iota`  THEN Auto) }


Latex:


Latex:
.....assertion..... 
1.  X  :  CubicalSet
2.  A  :  \{X  \mvdash{}  \_\}
3.  a  :  \{X  \mvdash{}  \_:A\}
4.  b  :  \{X  \mvdash{}  \_:A\}
5.  I  :  Cname  List
6.  alpha  :  X(I)
7.  z1  :  Cname
8.  \mneg{}(z1  \mmember{}  I)
9.  a1  :  A(iota(z1)(alpha))
10.  name-path-endpoints(X;A;a;b;I;alpha;z1;a1)
11.  z  :  Cname
12.  \mneg{}(z  \mmember{}  I)
13.  b2  :  named-path(X;A;a;b;I;alpha;z)
14.  (a1  iota(z1)(alpha)  rename-one-name(z1;z))  =  b2
15.  ((a1  iota(z1)(alpha)  rename-one-name(z1;z))  rename-one-name(z1;z)(iota(z1)(alpha))  ...)
=  (a1  iota(z1)(alpha)  (rename-one-name(z1;z)  o  rename-one-name(z;z1)))
16.  (rename-one-name(z1;z)  o  rename-one-name(z;z1))  =  1
\mvdash{}  (iota(z1)(alpha)  =  rename-one-name(z;z1)(iota(z)(alpha)))
\mwedge{}  (iota(z)(alpha)  =  rename-one-name(z1;z)(iota(z1)(alpha)))


By


Latex:
(D  0
  THEN  RWO  "cube-set-restriction-comp"  0
  THEN  Auto
  THEN  Symmetry
  THEN  BLemma  `rename-one-iota` 
  THEN  Auto)




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