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Step * 4 1 1 1 of Lemma free-dist-lattice-adjunction


1. ∀Y:Type. (Point(free-dl(Y)) ~ free-dl-type(Y))
2. ∀Y:Type. ∀y:Y.  (free-dl-generator(y) ∈ Point(free-dl(Y)))
3. b1 : BoundedDistributiveLattice
4. b2 : BoundedDistributiveLattice
5. g : Hom(b1;b2)
6. x : Point(b1)
7. (fdl-hom(free-dl(Point(b2));λx.free-dl-generator(g x)) free-dl-generator(x))
= free-dl-generator(g x)
∈ Point(free-dl(Point(b2)))
⊢ (g (fdl-hom(b1;λg.g) free-dl-generator(x)))
= (fdl-hom(b2;λg.g) (fdl-hom(free-dl(Point(b2));λx.free-dl-generator(g x)) free-dl-generator(x)))
∈ Point(b2)
BY
{ (ApFunToHypEquands `Z' ⌜fdl-hom(b2;λg.g) Z⌝ ⌜Point(b2)⌝ (-1)⋅ THENA Auto) }

1
1. ∀Y:Type. (Point(free-dl(Y)) ~ free-dl-type(Y))
2. ∀Y:Type. ∀y:Y.  (free-dl-generator(y) ∈ Point(free-dl(Y)))
3. b1 : BoundedDistributiveLattice
4. b2 : BoundedDistributiveLattice
5. g : Hom(b1;b2)
6. x : Point(b1)
7. (fdl-hom(free-dl(Point(b2));λx.free-dl-generator(g x)) free-dl-generator(x))
= free-dl-generator(g x)
∈ Point(free-dl(Point(b2)))
8. (fdl-hom(b2;λg.g) (fdl-hom(free-dl(Point(b2));λx.free-dl-generator(g x)) free-dl-generator(x)))
= (fdl-hom(b2;λg.g) free-dl-generator(g x))
∈ Point(b2)
⊢ (g (fdl-hom(b1;λg.g) free-dl-generator(x)))
= (fdl-hom(b2;λg.g) (fdl-hom(free-dl(Point(b2));λx.free-dl-generator(g x)) free-dl-generator(x)))
∈ Point(b2)

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Latex:

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Latex:

1.  \mforall{}Y:Type.  (Point(free-dl(Y))  \msim{}  free-dl-type(Y))
2.  \mforall{}Y:Type.  \mforall{}y:Y.    (free-dl-generator(y)  \mmember{}  Point(free-dl(Y)))
3.  b1  :  BoundedDistributiveLattice
4.  b2  :  BoundedDistributiveLattice
5.  g  :  Hom(b1;b2)
6.  x  :  Point(b1)
7.  (fdl-hom(free-dl(Point(b2));\mlambda{}x.free-dl-generator(g  x))  free-dl-generator(x))
=  free-dl-generator(g  x)
\mvdash{}  (g  (fdl-hom(b1;\mlambda{}g.g)  free-dl-generator(x)))
=  (fdl-hom(b2;\mlambda{}g.g)  (fdl-hom(free-dl(Point(b2));\mlambda{}x.free-dl-generator(g  x))  free-dl-generator(x)))


By

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Latex:
(ApFunToHypEquands  `Z'  \mkleeneopen{}fdl-hom(b2;\mlambda{}g.g)  Z\mkleeneclose{}  \mkleeneopen{}Point(b2)\mkleeneclose{}  (-1)\mcdot{}  THENA  Auto)

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