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Step * 1 2 of Lemma generated-subspace-is-least


1. K : Rng
2. vs : VectorSpace(K)
3. [P] : Point(vs) ⟶ ℙ
4. S : Point(vs) ⟶ ℙ
5. vs-subspace(K;vs;w.S[w])
6. ∀v:Point(vs). (P[v] ⇒ S[v])
7. v : Point(vs)
8. val : |K|
9. left_subtree : l_tree(x:Point(vs) × P[x];|K|)
10. right_subtree : l_tree(x:Point(vs) × P[x];|K|)
11. S[vs-tree-val(vs;left_subtree)]
12. S[vs-tree-val(vs;right_subtree)]
⊢ S[val * vs-tree-val(vs;left_subtree) + vs-tree-val(vs;right_subtree)]
BY
{ (Thin 6 THEN RepeatFor 2 (BackThruSomeHyp') THEN Auto) }

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Latex:

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Latex:

1.  K  :  Rng
2.  vs  :  VectorSpace(K)
3.  [P]  :  Point(vs)  {}\mrightarrow{}  \mBbbP{}
4.  S  :  Point(vs)  {}\mrightarrow{}  \mBbbP{}
5.  vs-subspace(K;vs;w.S[w])
6.  \mforall{}v:Point(vs).  (P[v]  {}\mRightarrow{}  S[v])
7.  v  :  Point(vs)
8.  val  :  |K|
9.  left$_{subtree}$  :  l\_tree(x:Point(vs)  \mtimes{}  P[x];|K|)
10.  right$_{subtree}$  :  l\_tree(x:Point(vs)  \mtimes{}  P[x];|K|)
11.  S[vs-tree-val(vs;left$_{subtree}$)]
12.  S[vs-tree-val(vs;right$_{subtree}$)]
\mvdash{}  S[val  *  vs-tree-val(vs;left$_{subtree}$)  +  vs-tree-val(vs;right$_{\000Csubtree}$)]


By

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Latex:
(Thin  6  THEN  RepeatFor  2  (BackThruSomeHyp')  THEN  Auto)

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