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Step * 2 1 of Lemma free-dlwc-basis

.....subterm..... T:t
2:n
1. T : Type
2. eq : EqDecider(T)
3. Cs : T ⟶ fset(fset(T))
4. x : Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
5. x = \/(λs.{s}"(x)) ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
⊢ λs./\(λx.free-dlwc-inc(eq;a.Cs[a];x)"(s))"(x)
= λs.{s}"(x)
∈ fset(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
BY
{ ((Assert deq-fset(deq-fset(eq)) ∈ EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))) BY
          (RWO "free-dlwc-point" 0 THEN Auto))
   THEN (Assert x ∈ fset(fset(T)) BY
               (RWO "free-dlwc-point" 4 THEN Auto))
   ) }

1
1. T : Type
2. eq : EqDecider(T)
3. Cs : T ⟶ fset(fset(T))
4. x : Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
5. x = \/(λs.{s}"(x)) ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
6. deq-fset(deq-fset(eq)) ∈ EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
7. x ∈ fset(fset(T))
⊢ λs./\(λx.free-dlwc-inc(eq;a.Cs[a];x)"(s))"(x)
= λs.{s}"(x)
∈ fset(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))

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Latex:

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Latex:
.....subterm.....  T:t
2:n
1.  T  :  Type
2.  eq  :  EqDecider(T)
3.  Cs  :  T  {}\mrightarrow{}  fset(fset(T))
4.  x  :  Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
5.  x  =  \mbackslash{}/(\mlambda{}s.\{s\}"(x))
\mvdash{}  \mlambda{}s./\mbackslash{}(\mlambda{}x.free-dlwc-inc(eq;a.Cs[a];x)"(s))"(x)  =  \mlambda{}s.\{s\}"(x)


By

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Latex:
((Assert  deq-fset(deq-fset(eq))
                  \mmember{}  EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))  BY
                (RWO  "free-dlwc-point"  0  THEN  Auto))
  THEN  (Assert  x  \mmember{}  fset(fset(T))  BY
                          (RWO  "free-dlwc-point"  4  THEN  Auto))
  )

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