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Step * of Lemma vs-bag-add-append

∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[S:Type]. ∀[f:S ⟶ Point(vs)]. ∀[bs,cs:bag(S)].
  (Σ{f[b] | b ∈ bs + cs} = Σ{f[b] | b ∈ bs} + Σ{f[b] | b ∈ cs} ∈ Point(vs))
BY
{ (Auto
   THEN Unfold `vs-bag-add` 0
   THEN (InstLemma `bag-summation-append` [⌜Point(vs)⌝;⌜λx,y. x + y⌝;⌜0⌝;⌜S⌝;⌜f⌝;⌜bs⌝;⌜cs⌝]⋅ THENM Reduce -1)
   THEN Auto
   THEN RepeatFor 2 ((D 0 THEN Reduce 0 THEN Auto))) }

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Latex:

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Latex:
\mforall{}[K:Rng].  \mforall{}[vs:VectorSpace(K)].  \mforall{}[S:Type].  \mforall{}[f:S  {}\mrightarrow{}  Point(vs)].  \mforall{}[bs,cs:bag(S)].
    (\mSigma{}\{f[b]  |  b  \mmember{}  bs  +  cs\}  =  \mSigma{}\{f[b]  |  b  \mmember{}  bs\}  +  \mSigma{}\{f[b]  |  b  \mmember{}  cs\})


By

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Latex:
(Auto
  THEN  Unfold  `vs-bag-add`  0
  THEN  (InstLemma  `bag-summation-append`  [\mkleeneopen{}Point(vs)\mkleeneclose{};\mkleeneopen{}\mlambda{}x,y.  x  +  y\mkleeneclose{};\mkleeneopen{}0\mkleeneclose{};\mkleeneopen{}S\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}bs\mkleeneclose{};\mkleeneopen{}cs\mkleeneclose{}]\mcdot{}
  THENM  Reduce  -1
  )
  THEN  Auto
  THEN  RepeatFor  2  ((D  0  THEN  Reduce  0  THEN  Auto)))

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