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Step * 1 1 1 of Lemma free-dl-le


1. [T] : Type
2. eq : EqDecider(T)@i
3. x : {ac:fset(fset(T))| ↑fset-antichain(eq;ac)} @i
4. y : {ac:fset(fset(T))| ↑fset-antichain(eq;ac)} @i
⊢ x = fset-ac-glb(eq;x;y) ∈ {ac:fset(fset(T))| ↑fset-antichain(eq;ac)}  ⇐⇒ fset-ac-le(eq;x;y)
BY
{ ((InstLemma `fset-ac-glb-is-glb` [⌜T⌝;⌜eq⌝;⌜x⌝;⌜y⌝]⋅ THENA Auto)
   THEN D -1
   THEN Auto
   THEN InstLemma `fset-ac-order` [⌜T⌝;⌜eq⌝]⋅
   THEN Auto
   THEN D -1
   THEN Auto) }

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Latex:

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Latex:

1.  [T]  :  Type
2.  eq  :  EqDecider(T)@i
3.  x  :  \{ac:fset(fset(T))|  \muparrow{}fset-antichain(eq;ac)\}  @i
4.  y  :  \{ac:fset(fset(T))|  \muparrow{}fset-antichain(eq;ac)\}  @i
\mvdash{}  x  =  fset-ac-glb(eq;x;y)  \mLeftarrow{}{}\mRightarrow{}  fset-ac-le(eq;x;y)


By

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Latex:
((InstLemma  `fset-ac-glb-is-glb`  [\mkleeneopen{}T\mkleeneclose{};\mkleeneopen{}eq\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{};\mkleeneopen{}y\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  D  -1
  THEN  Auto
  THEN  InstLemma  `fset-ac-order`  [\mkleeneopen{}T\mkleeneclose{};\mkleeneopen{}eq\mkleeneclose{}]\mcdot{}
  THEN  Auto
  THEN  D  -1
  THEN  Auto)

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