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Step * 2 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 : Point(free-DeMorgan-lattice(names(I);NamesDeq))
6. 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))
BY
{ ((Subst' dM-to-FL(I;x ∧ y) ~ (λz.dM-to-FL(I;z)) x ∧ y 0 THENA (Reduce 0 THEN Auto))
   THEN (Subst' dM-to-FL(I;x) ~ (λz.dM-to-FL(I;z)) x 0 THENA (Reduce 0 THEN Auto))
   THEN (Subst' dM-to-FL(I;y) ~ (λz.dM-to-FL(I;z)) y 0 THENA (Reduce 0 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 : Point(free-DeMorgan-lattice(names(I);NamesDeq))
6. y : Point(free-DeMorgan-lattice(names(I);NamesDeq))
⊢ ((λz.dM-to-FL(I;z)) x ∧ y) = (λz.dM-to-FL(I;z)) x ∧ (λz.dM-to-FL(I;z)) y ∈ 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.  x  :  Point(free-DeMorgan-lattice(names(I);NamesDeq))
6.  y  :  Point(free-DeMorgan-lattice(names(I);NamesDeq))
\mvdash{}  dM-to-FL(I;x  \mwedge{}  y)  =  dM-to-FL(I;x)  \mwedge{}  dM-to-FL(I;y)


By

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Latex:
((Subst'  dM-to-FL(I;x  \mwedge{}  y)  \msim{}  (\mlambda{}z.dM-to-FL(I;z))  x  \mwedge{}  y  0  THENA  (Reduce  0  THEN  Auto))
  THEN  (Subst'  dM-to-FL(I;x)  \msim{}  (\mlambda{}z.dM-to-FL(I;z))  x  0  THENA  (Reduce  0  THEN  Auto))
  THEN  (Subst'  dM-to-FL(I;y)  \msim{}  (\mlambda{}z.dM-to-FL(I;z))  y  0  THENA  (Reduce  0  THEN  Auto)))

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