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Step * 3 1 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. d : Type
4. x : d
5. (fdl-hom(free-dl(Point(free-dl(d)));λx.free-dl-generator(free-dl-generator(x))) free-dl-generator(x))
= free-dl-generator(free-dl-generator(x))
∈ Point(free-dl(Point(free-dl(d))))
⊢ (fdl-hom(free-dl(d);λg.g) 
   (fdl-hom(free-dl(Point(free-dl(d)));λx.free-dl-generator(free-dl-generator(x))) free-dl-generator(x)))
= free-dl-generator(x)
∈ Point(free-dl(d))
BY
{ (ApFunToHypEquands `Z' ⌜fdl-hom(free-dl(d);λg.g) Z⌝ ⌜Point(free-dl(d))⌝ (-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. d : Type
4. x : d
5. (fdl-hom(free-dl(Point(free-dl(d)));λx.free-dl-generator(free-dl-generator(x))) free-dl-generator(x))
= free-dl-generator(free-dl-generator(x))
∈ Point(free-dl(Point(free-dl(d))))
6. (fdl-hom(free-dl(d);λg.g) 
    (fdl-hom(free-dl(Point(free-dl(d)));λx.free-dl-generator(free-dl-generator(x))) free-dl-generator(x)))
= (fdl-hom(free-dl(d);λg.g) free-dl-generator(free-dl-generator(x)))
∈ Point(free-dl(d))
⊢ (fdl-hom(free-dl(d);λg.g) 
   (fdl-hom(free-dl(Point(free-dl(d)));λx.free-dl-generator(free-dl-generator(x))) free-dl-generator(x)))
= free-dl-generator(x)
∈ Point(free-dl(d))

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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.  d  :  Type
4.  x  :  d
5.  (fdl-hom(free-dl(Point(free-dl(d)));\mlambda{}x.free-dl-generator(free-dl-generator(x))) 
        free-dl-generator(x))
=  free-dl-generator(free-dl-generator(x))
\mvdash{}  (fdl-hom(free-dl(d);\mlambda{}g.g) 
      (fdl-hom(free-dl(Point(free-dl(d)));\mlambda{}x.free-dl-generator(free-dl-generator(x))) 
        free-dl-generator(x)))
=  free-dl-generator(x)


By

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

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