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Step * 2 1 1 1 of Lemma lattice-extend-meet

.....equality..... 
1. T : Type
2. eq : EqDecider(T)
3. L : BoundedDistributiveLattice
4. eqL : EqDecider(Point(L))
5. f : T ⟶ Point(L)
6. a : {ac:fset(fset(T))| ↑fset-antichain(eq;ac)} 
7. b : {ac:fset(fset(T))| ↑fset-antichain(eq;ac)} 
8. ∀[s:fset(fset(T))]. (λls./\(ls)"(λxs.f"(xs)"(s)) = λx./\(f"(x))"(s) ∈ fset(Point(L)))
⊢ λxs.f"(xs)"(f-union(deq-fset(eq);deq-fset(eq);a;as.λbs.as ⋃ bs"(b)))
= f-union(deq-fset(eqL);deq-fset(eqL);λxs.f"(xs)"(a);as.λbs.as ⋃ bs"(λxs.f"(xs)"(b)))
∈ fset(fset(Point(L)))
BY
{ ((Using [`eq',⌜deq-fset(eqL)⌝] (BLemma `fset-extensionality`)⋅ THENA Auto) THEN (D 0 THENA Auto)) }

1
1. T : Type
2. eq : EqDecider(T)
3. L : BoundedDistributiveLattice
4. eqL : EqDecider(Point(L))
5. f : T ⟶ Point(L)
6. a : {ac:fset(fset(T))| ↑fset-antichain(eq;ac)} 
7. b : {ac:fset(fset(T))| ↑fset-antichain(eq;ac)} 
8. ∀[s:fset(fset(T))]. (λls./\(ls)"(λxs.f"(xs)"(s)) = λx./\(f"(x))"(s) ∈ fset(Point(L)))
9. a1 : fset(Point(L))
⊢ uiff(a1 ∈ λxs.f"(xs)"
            (f-union(deq-fset(eq);deq-fset(eq);a;as.λbs.as ⋃ bs"
                                                    (b)));a1 ∈ f-union(deq-fset(eqL);deq-fset(eqL);λxs.f"(xs)"
                                                                                                   (a);as.λbs.as ⋃ bs"
                                                                                                          (λxs.f"(xs)"
                                                                                                           (b))))

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Latex:

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Latex:
.....equality..... 
1.  T  :  Type
2.  eq  :  EqDecider(T)
3.  L  :  BoundedDistributiveLattice
4.  eqL  :  EqDecider(Point(L))
5.  f  :  T  {}\mrightarrow{}  Point(L)
6.  a  :  \{ac:fset(fset(T))|  \muparrow{}fset-antichain(eq;ac)\} 
7.  b  :  \{ac:fset(fset(T))|  \muparrow{}fset-antichain(eq;ac)\} 
8.  \mforall{}[s:fset(fset(T))].  (\mlambda{}ls./\mbackslash{}(ls)"(\mlambda{}xs.f"(xs)"(s))  =  \mlambda{}x./\mbackslash{}(f"(x))"(s))
\mvdash{}  \mlambda{}xs.f"(xs)"(f-union(deq-fset(eq);deq-fset(eq);a;as.\mlambda{}bs.as  \mcup{}  bs"(b)))
=  f-union(deq-fset(eqL);deq-fset(eqL);\mlambda{}xs.f"(xs)"(a);as.\mlambda{}bs.as  \mcup{}  bs"(\mlambda{}xs.f"(xs)"(b)))


By

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Latex:
((Using  [`eq',\mkleeneopen{}deq-fset(eqL)\mkleeneclose{}]  (BLemma  `fset-extensionality`)\mcdot{}  THENA  Auto)  THEN  (D  0  THENA  Auto))

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