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Step * 1 of Lemma cubical-id-equiv-subset


1. G : Top
2. phi : Top
3. A : Top
⊢ (λcubical-pair((q)p @ q;path-contraction(G, phi.A.(A)p.(Path_((A)p)p (q)p q);q))) 
~ (λcubical-pair((q)p @ q;path-contraction(G.A.(A)p.(Path_((A)p)p (q)p q);q)))
BY
{ Subst' path-contraction(G, phi.A.(A)p.(Path_((A)p)p (q)p q);q) 
~ path-contraction(G.A.(A)p.(Path_((A)p)p (q)p q);q) 0 }

1
.....equality..... 
1. G : Top
2. phi : Top
3. A : Top
⊢ path-contraction(G, phi.A.(A)p.(Path_((A)p)p (q)p q);q) ~ path-contraction(G.A.(A)p.(Path_((A)p)p (q)p q);q)

2
1. G : Top
2. phi : Top
3. A : Top
⊢ (λcubical-pair((q)p @ q;path-contraction(G.A.(A)p.(Path_((A)p)p (q)p q);q))) 
~ (λcubical-pair((q)p @ q;path-contraction(G.A.(A)p.(Path_((A)p)p (q)p q);q)))

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Latex:

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Latex:

1.  G  :  Top
2.  phi  :  Top
3.  A  :  Top
\mvdash{}  (\mlambda{}cubical-pair((q)p  @  q;path-contraction(G,  phi.A.(A)p.(Path\_((A)p)p  (q)p  q);q))) 
\msim{}  (\mlambda{}cubical-pair((q)p  @  q;path-contraction(G.A.(A)p.(Path\_((A)p)p  (q)p  q);q)))


By

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Latex:
Subst'  path-contraction(G,  phi.A.(A)p.(Path\_((A)p)p  (q)p  q);q) 
\msim{}  path-contraction(G.A.(A)p.(Path\_((A)p)p  (q)p  q);q)  0

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