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Step * 2 1 1 of Lemma partition-sum-bound


1. I : Interval
2. icompact(I)
3. f : I ⟶ℝ
4. mc : f[x] continuous for x ∈ I
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} )
9. ||full-partition(I;p)|| = (||p|| + 2) ∈ ℤ
⊢ Σ{|(f (a i)) * (full-partition(I;p)[i + 1] - full-partition(I;p)[i])| | 0≤i≤||full-partition(I;p)|| - 2} ≤ (||f[x]||_I
* |I|)
BY
{ Assert ⌜Σ{|(f (a 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;p)[i + 1] - full-partition(I;p)[i]) | 0≤i≤||full-partition(I;p)|| 
          - 2}⌝⋅ }

1
.....assertion..... 
1. I : Interval
2. icompact(I)
3. f : I ⟶ℝ
4. mc : f[x] continuous for x ∈ I
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} )
9. ||full-partition(I;p)|| = (||p|| + 2) ∈ ℤ
⊢ Σ{|(f (a 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;p)[i + 1] - full-partition(I;p)[i]) | 0≤i≤||full-partition(I;p)|| - 2}

2
1. I : Interval
2. icompact(I)
3. f : I ⟶ℝ
4. mc : f[x] continuous for x ∈ I
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} )
9. ||full-partition(I;p)|| = (||p|| + 2) ∈ ℤ
10. Σ{|(f (a 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;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} ≤ (||f[x]||_I
* |I|)

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Latex:

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Latex:

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


By

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Latex:
Assert  \mkleeneopen{}\mSigma{}\{|(f  (a  i))
                *  (full-partition(I;p)[i  +  1]  -  full-partition(I;p)[i])|  |  0\mleq{}i\mleq{}||full-partition(I;p)|| 
                -  2\}  \mleq{}  \mSigma{}\{||f[x]||\_I
                *  (full-partition(I;p)[i  +  1]  -  full-partition(I;p)[i])  |  0\mleq{}i\mleq{}||full-partition(I;p)||  -  2\}\mkleeneclose{}\mcdot{}

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