---

Step * of Lemma do-apply-p-co-restrict

∀[A,B:Type]. ∀[f:A ⟶ (B + Top)]. ∀[P:A ⟶ ℙ]. ∀[p:∀x:A. Dec(P[x])]. ∀[x:A].
  do-apply(p-co-restrict(f;p);x) = do-apply(f;x) ∈ B supposing ↑can-apply(p-co-restrict(f;p);x)
BY
{ (Unfold `p-co-restrict` 0
   THEN Auto
   THEN RWO "do-apply-compose" 0
   THEN Auto
   THEN (RWO "can-apply-compose-iff" (-1))
   THEN Auto
   THEN RWO "do-apply-p-co-filter" 0
   THEN Auto
   THEN (All (RWO "do-apply-p-co-filter"))
   THEN Auto) }

---

Latex:

---

Latex:
\mforall{}[A,B:Type].  \mforall{}[f:A  {}\mrightarrow{}  (B  +  Top)].  \mforall{}[P:A  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[p:\mforall{}x:A.  Dec(P[x])].  \mforall{}[x:A].
    do-apply(p-co-restrict(f;p);x)  =  do-apply(f;x)  supposing  \muparrow{}can-apply(p-co-restrict(f;p);x)


By

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Latex:
(Unfold  `p-co-restrict`  0
  THEN  Auto
  THEN  RWO  "do-apply-compose"  0
  THEN  Auto
  THEN  (RWO  "can-apply-compose-iff"  (-1))
  THEN  Auto
  THEN  RWO  "do-apply-p-co-filter"  0
  THEN  Auto
  THEN  (All  (RWO  "do-apply-p-co-filter"))
  THEN  Auto)

---

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