Step * 1 of Lemma univ-trans-equiv_path

.....assertion..... 
1. CubicalSet{j}
2. {G ⊢ _:c𝕌}
3. {G ⊢ _:c𝕌}
4. {G ⊢ _:Equiv(decode(A);decode(B))}
5. equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f) ∈ G.𝕀 +⊢ Compositon(equiv-path1(G;decode(A);decode(B);f))
6. cfun-to-cop(G.𝕀;equiv-path1(G;decode(A);decode(B);f);equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))
   ∈ G.𝕀 ⊢ CompOp(equiv-path1(G;decode(A);decode(B);f))
7. decode(encode(equiv-path1(G;decode(A);decode(B);f);cfun-to-cop(G.𝕀;equiv-path1(G;decode(A);decode(B);f)
                                                          ;equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))))
equiv-path1(G;decode(A);decode(B);f)
∈ {G.𝕀 ⊢ _}
8. compOp(encode(equiv-path1(G;decode(A);decode(B);f);cfun-to-cop(G.𝕀;equiv-path1(G;decode(A);decode(B);f)
                                                          ;equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))))
cfun-to-cop(G.𝕀;equiv-path1(G;decode(A);decode(B);f);equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))
∈ G.𝕀 ⊢ CompOp(equiv-path1(G;decode(A);decode(B);f))
⊢ transprt-fun(G;equiv-path1(G;decode(A);decode(B);f);equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))
trans-equiv-path(G;A;B;f)
∈ {G ⊢ _:(decode(A) ⟶ decode(B))}
BY
(RepeatFor (Thin (-1))
   THEN RepUR ``transprt-fun`` 0
   THEN Assert ⌜transprt(G.decode(A);(equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))p+;q))
                trans-equiv-path(G;A;B;f)
                ∈ {G ⊢ _:(decode(A) ⟶ decode(B))}⌝⋅}

1
.....assertion..... 
1. CubicalSet{j}
2. {G ⊢ _:c𝕌}
3. {G ⊢ _:c𝕌}
4. {G ⊢ _:Equiv(decode(A);decode(B))}
5. equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f) ∈ G.𝕀 +⊢ Compositon(equiv-path1(G;decode(A);decode(B);f))
6. cfun-to-cop(G.𝕀;equiv-path1(G;decode(A);decode(B);f);equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))
   ∈ G.𝕀 ⊢ CompOp(equiv-path1(G;decode(A);decode(B);f))
⊢ transprt(G.decode(A);(equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))p+;q))
trans-equiv-path(G;A;B;f)
∈ {G ⊢ _:(decode(A) ⟶ decode(B))}

2
1. CubicalSet{j}
2. {G ⊢ _:c𝕌}
3. {G ⊢ _:c𝕌}
4. {G ⊢ _:Equiv(decode(A);decode(B))}
5. equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f) ∈ G.𝕀 +⊢ Compositon(equiv-path1(G;decode(A);decode(B);f))
6. cfun-to-cop(G.𝕀;equiv-path1(G;decode(A);decode(B);f);equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))
   ∈ G.𝕀 ⊢ CompOp(equiv-path1(G;decode(A);decode(B);f))
7. transprt(G.decode(A);(equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))p+;q))
trans-equiv-path(G;A;B;f)
∈ {G ⊢ _:(decode(A) ⟶ decode(B))}
⊢ transprt(G.(equiv-path1(G;decode(A);decode(B);f))
                [0(𝕀)];(equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))p+;q))
trans-equiv-path(G;A;B;f)
∈ {G ⊢ _:(decode(A) ⟶ decode(B))}


Latex:


Latex:
.....assertion..... 
1.  G  :  CubicalSet\{j\}
2.  A  :  \{G  \mvdash{}  \_:c\mBbbU{}\}
3.  B  :  \{G  \mvdash{}  \_:c\mBbbU{}\}
4.  f  :  \{G  \mvdash{}  \_:Equiv(decode(A);decode(B))\}
5.  equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f)
      \mmember{}  G.\mBbbI{}  +\mvdash{}  Compositon(equiv-path1(G;decode(A);decode(B);f))
6.  cfun-to-cop(G.\mBbbI{};equiv-path1(G;decode(A);decode(B);f)
              ;equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))
      \mmember{}  G.\mBbbI{}  \mvdash{}  CompOp(equiv-path1(G;decode(A);decode(B);f))
7.  decode(encode(equiv-path1(G;decode(A);decode(B);f);
                                  cfun-to-cop(G.\mBbbI{};equiv-path1(G;decode(A);decode(B);f)
                                          ;equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))))
=  equiv-path1(G;decode(A);decode(B);f)
8.  compOp(encode(equiv-path1(G;decode(A);decode(B);f);
                                  cfun-to-cop(G.\mBbbI{};equiv-path1(G;decode(A);decode(B);f)
                                          ;equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))))
=  cfun-to-cop(G.\mBbbI{};equiv-path1(G;decode(A);decode(B);f)
            ;equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))
\mvdash{}  transprt-fun(G;equiv-path1(G;decode(A);decode(B);f);equiv-path2(G;decode(A);decode(B);...;...;f))
=  trans-equiv-path(G;A;B;f)


By


Latex:
(RepeatFor  2  (Thin  (-1))
  THEN  RepUR  ``transprt-fun``  0
  THEN  Assert 
  \mkleeneopen{}(\mlambda{}transprt(G.decode(A);(equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))p+;q))
    =  trans-equiv-path(G;A;B;f)\mkleeneclose{}\mcdot{})




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