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Step * 1 of Lemma partition-sum_functionality


1. I : Interval
2. icompact(I)
3. p : partition(I)
4. q : ℝ List
5. ||q|| = ||p|| ∈ ℤ
6. ∀i:ℕ||q||. (q[i] = p[i])
7. f : I ⟶ℝ
8. x : partition-choice(full-partition(I;p))
9. ||full-partition(I;q)|| = ||full-partition(I;p)|| ∈ ℤ
⊢ Σ{(f (x i)) * (full-partition(I;p)[i + 1] - full-partition(I;p)[i]) | 0≤i≤||full-partition(I;p)|| - 2}
= Σ{(f (x i)) * (full-partition(I;q)[i + 1] - full-partition(I;q)[i]) | 0≤i≤||full-partition(I;p)|| - 2}
BY
{ xxx(Assert ∀i:ℕ. ((i ≤ (||full-partition(I;p)|| - 1)) ⇒ (full-partition(I;q)[i] ∈ ℝ)) BY
            ((UnivCD THENA Auto)
             THEN RevHypSubst' (-3) (-1)
             THEN Thin (-3)
             THEN RepUR ``full-partition`` -1
             THEN RepUR ``full-partition`` 0
             THEN Auto'))xxx }

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

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Latex:

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Latex:

1.  I  :  Interval
2.  icompact(I)
3.  p  :  partition(I)
4.  q  :  \mBbbR{}  List
5.  ||q||  =  ||p||
6.  \mforall{}i:\mBbbN{}||q||.  (q[i]  =  p[i])
7.  f  :  I  {}\mrightarrow{}\mBbbR{}
8.  x  :  partition-choice(full-partition(I;p))
9.  ||full-partition(I;q)||  =  ||full-partition(I;p)||
\mvdash{}  \mSigma{}\{(f  (x  i))  *  (full-partition(I;p)[i  +  1]  -  full-partition(I;p)[i])  |  0\mleq{}i\mleq{}||full-partition(I;p)|| 
-  2\}
=  \mSigma{}\{(f  (x  i))  *  (full-partition(I;q)[i  +  1]  -  full-partition(I;q)[i])  |  0\mleq{}i\mleq{}||full-partition(I;p)|| 
    -  2\}


By

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Latex:
xxx(Assert  \mforall{}i:\mBbbN{}.  ((i  \mleq{}  (||full-partition(I;p)||  -  1))  {}\mRightarrow{}  (full-partition(I;q)[i]  \mmember{}  \mBbbR{}))  BY
                    ((UnivCD  THENA  Auto)
                      THEN  RevHypSubst'  (-3)  (-1)
                      THEN  Thin  (-3)
                      THEN  RepUR  ``full-partition``  -1
                      THEN  RepUR  ``full-partition``  0
                      THEN  Auto'))xxx

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