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Step * 1 2 1 1 2 2 1 1 of Lemma name-morph-decomp

.....equality..... 
1. I : Cname List
2. J : Cname List
3. f : name-morph(I;J)
4. ∀[f@0:{x:nameset(I)| ↑¬bisname(f x)}  ⟶ (Cname × ℕ2)]. (mapfilter(f@0;λx.(¬bisname(f x));I) ∈ (Cname × ℕ2) List)
5. mapfilter(λx.<x, f x>;λx.(¬bisname(f x));I) ∈ (Cname × ℕ2) List
6. nameset(I) ≡ nameset(filter(λx.(¬bisname(f x));I) @ filter(λx.isname(f x);I))
7. K : Cname List
8. nameset(I) ≡ nameset(filter(λx.(¬bisname(f x));I) @ K)
9. g : nameset(K) ⟶ nameset(J)
10. Inj(nameset(K);nameset(J);g)
11. face-maps-comp(mapfilter(λx.<x, f x>;λx.(¬bisname(f x));I)) ∈ name-morph(map(λp.(fst(p));mapfilter(λx.<x, f x>;
                                                                                                       λx.(¬bisname(f 
                                                                                                                    x));
                                                                                                       I))
    @ K;K)
⊢ map(λp.(fst(p));mapfilter(λx.<x, f x>;λx.(¬bisname(f x));I)) ~ filter(λx.(¬bisname(f x));I)
BY
{ (RepUR ``mapfilter`` 0
   THEN (RWO "map-map" 0 THENA Auto)
   THEN RepUR ``compose`` 0
   THEN RWO "map-id" 0
   THEN Try ((BLemma `filter_wf_top`  THEN Try (Fold `nameset` 0) THEN Auto))
   THEN Auto) }

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Latex:

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Latex:
.....equality..... 
1.  I  :  Cname  List
2.  J  :  Cname  List
3.  f  :  name-morph(I;J)
4.  \mforall{}[f@0:\{x:nameset(I)|  \muparrow{}\mneg{}\msubb{}isname(f  x)\}    {}\mrightarrow{}  (Cname  \mtimes{}  \mBbbN{}2)]
          (mapfilter(f@0;\mlambda{}x.(\mneg{}\msubb{}isname(f  x));I)  \mmember{}  (Cname  \mtimes{}  \mBbbN{}2)  List)
5.  mapfilter(\mlambda{}x.<x,  f  x>\mlambda{}x.(\mneg{}\msubb{}isname(f  x));I)  \mmember{}  (Cname  \mtimes{}  \mBbbN{}2)  List
6.  nameset(I)  \mequiv{}  nameset(filter(\mlambda{}x.(\mneg{}\msubb{}isname(f  x));I)  @  filter(\mlambda{}x.isname(f  x);I))
7.  K  :  Cname  List
8.  nameset(I)  \mequiv{}  nameset(filter(\mlambda{}x.(\mneg{}\msubb{}isname(f  x));I)  @  K)
9.  g  :  nameset(K)  {}\mrightarrow{}  nameset(J)
10.  Inj(nameset(K);nameset(J);g)
11.  face-maps-comp(mapfilter(\mlambda{}x.<x,  f  x>\mlambda{}x.(\mneg{}\msubb{}isname(f  x));I))
        \mmember{}  name-morph(map(\mlambda{}p.(fst(p));mapfilter(\mlambda{}x.<x,  f  x>\mlambda{}x.(\mneg{}\msubb{}isname(f  x));I))  @  K;K)
\mvdash{}  map(\mlambda{}p.(fst(p));mapfilter(\mlambda{}x.<x,  f  x>\mlambda{}x.(\mneg{}\msubb{}isname(f  x));I))  \msim{}  filter(\mlambda{}x.(\mneg{}\msubb{}isname(f  x));I)


By

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Latex:
(RepUR  ``mapfilter``  0
  THEN  (RWO  "map-map"  0  THENA  Auto)
  THEN  RepUR  ``compose``  0
  THEN  RWO  "map-id"  0
  THEN  Try  ((BLemma  `filter\_wf\_top`    THEN  Try  (Fold  `nameset`  0)  THEN  Auto))
  THEN  Auto)

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