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Step * 1 1 of Lemma dma-lift-compose-assoc


1. I : Type
2. J : Type
3. K : Type
4. H : Type
5. eqi : EqDecider(I)
6. eqj : EqDecider(J)
7. eqk : EqDecider(K)
8. f : J ⟶ Point(free-DeMorgan-algebra(I;eqi))
9. g : K ⟶ Point(free-DeMorgan-algebra(J;eqj))
10. h : H ⟶ Point(free-DeMorgan-algebra(K;eqk))
11. x : H
⊢ (free-dma-lift(K;eqk;free-DeMorgan-algebra(I;eqi);free-dml-deq(I;eqi);free-dma-lift(J;eqj;...;free-dml-deq(I;eqi);f)
    o g) 
   (h x))
= ((free-dma-lift(J;eqj;free-DeMorgan-algebra(I;eqi);free-dml-deq(I;eqi);f)
    o free-dma-lift(K;eqk;free-DeMorgan-algebra(J;eqj);free-dml-deq(J;eqj);g)) 
   (h x))
∈ Point(free-DeMorgan-algebra(I;eqi))
BY
{ Assert ⌜∀X:Type. ∀eq:EqDecider(X).  (free-dml-deq(X;eq) ∈ EqDecider(Point(free-DeMorgan-algebra(X;eq))))⌝⋅ }

1
.....assertion..... 
1. I : Type
2. J : Type
3. K : Type
4. H : Type
5. eqi : EqDecider(I)
6. eqj : EqDecider(J)
7. eqk : EqDecider(K)
8. f : J ⟶ Point(free-DeMorgan-algebra(I;eqi))
9. g : K ⟶ Point(free-DeMorgan-algebra(J;eqj))
10. h : H ⟶ Point(free-DeMorgan-algebra(K;eqk))
11. x : H
⊢ ∀X:Type. ∀eq:EqDecider(X).  (free-dml-deq(X;eq) ∈ EqDecider(Point(free-DeMorgan-algebra(X;eq))))

2
1. I : Type
2. J : Type
3. K : Type
4. H : Type
5. eqi : EqDecider(I)
6. eqj : EqDecider(J)
7. eqk : EqDecider(K)
8. f : J ⟶ Point(free-DeMorgan-algebra(I;eqi))
9. g : K ⟶ Point(free-DeMorgan-algebra(J;eqj))
10. h : H ⟶ Point(free-DeMorgan-algebra(K;eqk))
11. x : H
12. ∀X:Type. ∀eq:EqDecider(X).  (free-dml-deq(X;eq) ∈ EqDecider(Point(free-DeMorgan-algebra(X;eq))))
⊢ (free-dma-lift(K;eqk;free-DeMorgan-algebra(I;eqi);free-dml-deq(I;eqi);free-dma-lift(J;eqj;...;free-dml-deq(I;eqi);f)
    o g) 
   (h x))
= ((free-dma-lift(J;eqj;free-DeMorgan-algebra(I;eqi);free-dml-deq(I;eqi);f)
    o free-dma-lift(K;eqk;free-DeMorgan-algebra(J;eqj);free-dml-deq(J;eqj);g)) 
   (h x))
∈ Point(free-DeMorgan-algebra(I;eqi))

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Latex:

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Latex:

1.  I  :  Type
2.  J  :  Type
3.  K  :  Type
4.  H  :  Type
5.  eqi  :  EqDecider(I)
6.  eqj  :  EqDecider(J)
7.  eqk  :  EqDecider(K)
8.  f  :  J  {}\mrightarrow{}  Point(free-DeMorgan-algebra(I;eqi))
9.  g  :  K  {}\mrightarrow{}  Point(free-DeMorgan-algebra(J;eqj))
10.  h  :  H  {}\mrightarrow{}  Point(free-DeMorgan-algebra(K;eqk))
11.  x  :  H
\mvdash{}  (free-dma-lift(K;eqk;free-DeMorgan-algebra(I;eqi);free-dml-deq(I;eqi);...  o  g) 
      (h  x))
=  ((free-dma-lift(J;eqj;free-DeMorgan-algebra(I;eqi);free-dml-deq(I;eqi);f)
        o  free-dma-lift(K;eqk;free-DeMorgan-algebra(J;eqj);free-dml-deq(J;eqj);g)) 
      (h  x))


By

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Latex:
Assert  \mkleeneopen{}\mforall{}X:Type.  \mforall{}eq:EqDecider(X).
                    (free-dml-deq(X;eq)  \mmember{}  EqDecider(Point(free-DeMorgan-algebra(X;eq))))\mkleeneclose{}\mcdot{}

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