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Step * 1 1 1 2 1 of Lemma free-dlwc-basis

.....assertion..... 
1. T : Type
2. eq : EqDecider(T)
3. Cs : T ⟶ fset(fset(T))
4. x : fset(fset(T))
5. ↑fset-antichain(eq;x)
6. fset-all(x;a.fset-contains-none(eq;a;x.Cs[x]))
7. x ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
8. deq-fset(deq-fset(eq)) ∈ EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
9. ∀s:fset(T). (s ∈ x ⇒ ({s} ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))))
10. λs.{s}"(x) ∈ fset(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
11. ∀[x@0:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))]. x@0 ≤ \/(λs.{s}"(x)) supposing x@0 ∈ λs.{s}"(x)
12. ∀[u:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))]
      ((∀x@0:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])). (x@0 ∈ λs.{s}"(x) ⇒ x@0 ≤ u))
      ⇒ \/(λs.{s}"(x)) ≤ u)
13. \/(λs.{s}"(x)) ≤ x
⊢ x ≤ \/(λs.{s}"(x))
BY
{ ((Assert \/(λs.{s}"(x)) ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])) BY
          Auto)
   THEN DupHyp (-1)
   THEN (RWO  "free-dlwc-point" (-1) THENA Auto)) }

1
1. T : Type
2. eq : EqDecider(T)
3. Cs : T ⟶ fset(fset(T))
4. x : fset(fset(T))
5. ↑fset-antichain(eq;x)
6. fset-all(x;a.fset-contains-none(eq;a;x.Cs[x]))
7. x ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
8. deq-fset(deq-fset(eq)) ∈ EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
9. ∀s:fset(T). (s ∈ x ⇒ ({s} ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))))
10. λs.{s}"(x) ∈ fset(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
11. ∀[x@0:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))]. x@0 ≤ \/(λs.{s}"(x)) supposing x@0 ∈ λs.{s}"(x)
12. ∀[u:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))]
      ((∀x@0:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])). (x@0 ∈ λs.{s}"(x) ⇒ x@0 ≤ u))
      ⇒ \/(λs.{s}"(x)) ≤ u)
13. \/(λs.{s}"(x)) ≤ x
14. \/(λs.{s}"(x)) ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
15. \/(λs.{s}"(x)) ∈ {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))} 
⊢ x ≤ \/(λs.{s}"(x))

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Latex:

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Latex:
.....assertion..... 
1.  T  :  Type
2.  eq  :  EqDecider(T)
3.  Cs  :  T  {}\mrightarrow{}  fset(fset(T))
4.  x  :  fset(fset(T))
5.  \muparrow{}fset-antichain(eq;x)
6.  fset-all(x;a.fset-contains-none(eq;a;x.Cs[x]))
7.  x  \mmember{}  Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
8.  deq-fset(deq-fset(eq))  \mmember{}  EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
9.  \mforall{}s:fset(T).  (s  \mmember{}  x  {}\mRightarrow{}  (\{s\}  \mmember{}  Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))))
10.  \mlambda{}s.\{s\}"(x)  \mmember{}  fset(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
11.  \mforall{}[x@0:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))]
            x@0  \mleq{}  \mbackslash{}/(\mlambda{}s.\{s\}"(x))  supposing  x@0  \mmember{}  \mlambda{}s.\{s\}"(x)
12.  \mforall{}[u:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))]
            ((\mforall{}x@0:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])).  (x@0  \mmember{}  \mlambda{}s.\{s\}"(x)  {}\mRightarrow{}  x@0  \mleq{}  u))
            {}\mRightarrow{}  \mbackslash{}/(\mlambda{}s.\{s\}"(x))  \mleq{}  u)
13.  \mbackslash{}/(\mlambda{}s.\{s\}"(x))  \mleq{}  x
\mvdash{}  x  \mleq{}  \mbackslash{}/(\mlambda{}s.\{s\}"(x))


By

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Latex:
((Assert  \mbackslash{}/(\mlambda{}s.\{s\}"(x))  \mmember{}  Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))  BY
                Auto)
  THEN  DupHyp  (-1)
  THEN  (RWO    "free-dlwc-point"  (-1)  THENA  Auto))

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