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Step * 3 of Lemma dM-to-FL-properties


1. ∀[I:fset(ℕ)]. (λz.dM-to-FL(I;z) ∈ Hom(free-DeMorgan-lattice(names(I);NamesDeq);face_lattice(I)))
2. I : fset(ℕ)
3. λz.dM-to-FL(I;z) ∈ Hom(free-DeMorgan-lattice(names(I);NamesDeq);face_lattice(I))
4. ∀x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
     (dM-to-FL(I;x ∨ y) = dM-to-FL(I;x) ∨ dM-to-FL(I;y) ∈ Point(face_lattice(I)))
5. ∀x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
     (dM-to-FL(I;x ∧ y) = dM-to-FL(I;x) ∧ dM-to-FL(I;y) ∈ Point(face_lattice(I)))
6. dM-to-FL(I;0) = 0 ∈ Point(face_lattice(I))
⊢ dM-to-FL(I;1) = 1 ∈ Point(face_lattice(I))
BY
{ ((Subst' dM-to-FL(I;1) ~ (λz.dM-to-FL(I;z)) 1 0 THENA (Reduce 0 THEN Auto))
   THEN GenConclTerm ⌜λz.dM-to-FL(I;z)⌝⋅
   THEN Auto) }

1
1. ∀[I:fset(ℕ)]. (λz.dM-to-FL(I;z) ∈ Hom(free-DeMorgan-lattice(names(I);NamesDeq);face_lattice(I)))
2. I : fset(ℕ)
3. λz.dM-to-FL(I;z) ∈ Hom(free-DeMorgan-lattice(names(I);NamesDeq);face_lattice(I))
4. ∀x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
     (dM-to-FL(I;x ∨ y) = dM-to-FL(I;x) ∨ dM-to-FL(I;y) ∈ Point(face_lattice(I)))
5. ∀x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
     (dM-to-FL(I;x ∧ y) = dM-to-FL(I;x) ∧ dM-to-FL(I;y) ∈ Point(face_lattice(I)))
6. dM-to-FL(I;0) = 0 ∈ Point(face_lattice(I))
7. v : Hom(free-DeMorgan-lattice(names(I);NamesDeq);face_lattice(I))
8. (λz.dM-to-FL(I;z)) = v ∈ Hom(free-DeMorgan-lattice(names(I);NamesDeq);face_lattice(I))
⊢ (v 1) = 1 ∈ Point(face_lattice(I))

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Latex:

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Latex:

1.  \mforall{}[I:fset(\mBbbN{})].  (\mlambda{}z.dM-to-FL(I;z)  \mmember{}  Hom(free-DeMorgan-lattice(names(I);NamesDeq);face\_lattice(I)))
2.  I  :  fset(\mBbbN{})
3.  \mlambda{}z.dM-to-FL(I;z)  \mmember{}  Hom(free-DeMorgan-lattice(names(I);NamesDeq);face\_lattice(I))
4.  \mforall{}x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
          (dM-to-FL(I;x  \mvee{}  y)  =  dM-to-FL(I;x)  \mvee{}  dM-to-FL(I;y))
5.  \mforall{}x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
          (dM-to-FL(I;x  \mwedge{}  y)  =  dM-to-FL(I;x)  \mwedge{}  dM-to-FL(I;y))
6.  dM-to-FL(I;0)  =  0
\mvdash{}  dM-to-FL(I;1)  =  1


By

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Latex:
((Subst'  dM-to-FL(I;1)  \msim{}  (\mlambda{}z.dM-to-FL(I;z))  1  0  THENA  (Reduce  0  THEN  Auto))
  THEN  GenConclTerm  \mkleeneopen{}\mlambda{}z.dM-to-FL(I;z)\mkleeneclose{}\mcdot{}
  THEN  Auto)

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