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Step * of Lemma partition-sum-radd

∀I:Interval
  (icompact(I)
  ⇒ (∀f,g:I ⟶ℝ. ∀p:partition(I). ∀y:partition-choice(full-partition(I;p)).
        (S(λx.((f x) + (g x));full-partition(I;p)) = (S(f;full-partition(I;p)) + S(g;full-partition(I;p))))))
BY
{ (Auto
   THEN Unfold `partition-sum` 0
   THEN Reduce 0
   THEN (GenConcl ⌜y = a ∈ (ℕ||p|| + 1 ⟶ {x:ℝ| x ∈ I} )⌝⋅ THENA Auto)) }

1
1. I : Interval
2. icompact(I)
3. f : I ⟶ℝ
4. g : I ⟶ℝ
5. p : partition(I)
6. y : partition-choice(full-partition(I;p))
7. a : ℕ||p|| + 1 ⟶ {x:ℝ| x ∈ I} 
8. y = a ∈ (ℕ||p|| + 1 ⟶ {x:ℝ| x ∈ I} )
⊢ Σ{((f (a i)) + (g (a i))) * (full-partition(I;p)[i + 1] - full-partition(I;p)[i]) | 0≤i≤||full-partition(I;p)|| - 2}
= (Σ{(f (a i)) * (full-partition(I;p)[i + 1] - full-partition(I;p)[i]) | 0≤i≤||full-partition(I;p)|| - 2}
  + Σ{(g (a i)) * (full-partition(I;p)[i + 1] - full-partition(I;p)[i]) | 0≤i≤||full-partition(I;p)|| - 2})

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Latex:

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Latex:
\mforall{}I:Interval
    (icompact(I)
    {}\mRightarrow{}  (\mforall{}f,g:I  {}\mrightarrow{}\mBbbR{}.  \mforall{}p:partition(I).  \mforall{}y:partition-choice(full-partition(I;p)).
                (S(\mlambda{}x.((f  x)  +  (g  x));full-partition(I;p))
                =  (S(f;full-partition(I;p))  +  S(g;full-partition(I;p))))))


By

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Latex:
(Auto  THEN  Unfold  `partition-sum`  0  THEN  Reduce  0  THEN  (GenConcl  \mkleeneopen{}y  =  a\mkleeneclose{}\mcdot{}  THENA  Auto))

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