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Step * 1 of Lemma face-lattice-basis


1. T : Type
2. eq : EqDecider(T)
3. x : fset(fset(T + T))
4. ↑fset-antichain(union-deq(T;T;eq;eq);x)
5. fset-all(x;a.fset-contains-none(union-deq(T;T;eq;eq);a;x.face-lattice-constraints(x)))
6. s : fset(T + T)
7. u : T + T
8. ↑fset-null({c ∈ face-lattice-constraints(u) | deq-f-subset(union-deq(T;T;eq;eq)) c {u}})
⊢ {{u}}
= {{u}}
∈ {ac:fset(fset(T + T))| 
   (↑fset-antichain(union-deq(T;T;eq;eq);ac))
   ∧ fset-all(ac;a.fset-contains-none(union-deq(T;T;eq;eq);a;x.face-lattice-constraints(x)))} 
BY
{ ((RWO "fl-point-sq<" 0 THENA Auto) THEN Fold `member` 0) }

1
1. T : Type
2. eq : EqDecider(T)
3. x : fset(fset(T + T))
4. ↑fset-antichain(union-deq(T;T;eq;eq);x)
5. fset-all(x;a.fset-contains-none(union-deq(T;T;eq;eq);a;x.face-lattice-constraints(x)))
6. s : fset(T + T)
7. u : T + T
8. ↑fset-null({c ∈ face-lattice-constraints(u) | deq-f-subset(union-deq(T;T;eq;eq)) c {u}})
⊢ {{u}} ∈ Point(face-lattice(T;eq))

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Latex:

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Latex:

1.  T  :  Type
2.  eq  :  EqDecider(T)
3.  x  :  fset(fset(T  +  T))
4.  \muparrow{}fset-antichain(union-deq(T;T;eq;eq);x)
5.  fset-all(x;a.fset-contains-none(union-deq(T;T;eq;eq);a;x.face-lattice-constraints(x)))
6.  s  :  fset(T  +  T)
7.  u  :  T  +  T
8.  \muparrow{}fset-null(\{c  \mmember{}  face-lattice-constraints(u)  |  deq-f-subset(union-deq(T;T;eq;eq))  c  \{u\}\})
\mvdash{}  \{\{u\}\}  =  \{\{u\}\}


By

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Latex:
((RWO  "fl-point-sq<"  0  THENA  Auto)  THEN  Fold  `member`  0)

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