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Step * 1 of Lemma integral-additive


1. [a] : ℝ
2. [b] : ℝ
3. [c] : ℝ
4. rmin(a;rmin(b;c)) ≤ rmax(a;rmax(b;c))
5. [f] : {f:[rmin(a;rmin(b;c)), rmax(a;rmax(b;c))] ⟶ℝ| ifun(f;[rmin(a;rmin(b;c)), rmax(a;rmax(b;c))])} 
⊢ a_∫-b f[x] dx = (a_∫-c f[x] dx + c_∫-b f[x] dx)
BY
{ ((InstLemma `integral-additive-lemma`    [⌜rmin(a;rmin(b;c))⌝;⌜rmax(a;rmax(b;c))⌝;⌜f⌝]⋅ THENA Auto)
   THEN (RWO "-1" 0 THENA Auto)
   ) }

1
1. a : ℝ
2. b : ℝ
3. c : ℝ
4. rmin(a;rmin(b;c)) ≤ rmax(a;rmax(b;c))
5. f : {f:[rmin(a;rmin(b;c)), rmax(a;rmax(b;c))] ⟶ℝ| ifun(f;[rmin(a;rmin(b;c)), rmax(a;rmax(b;c))])} 
6. ∀[a@0,b@0:{x:ℝ| x ∈ [rmin(a;rmin(b;c)), rmax(a;rmax(b;c))]} ].
     (a@0_∫-b@0 f[x] dx = (∫ f[x] dx on [rmin(a;rmin(b;c)), b@0] - ∫ f[x] dx on [rmin(a;rmin(b;c)), a@0]))
⊢ (∫ f[x] dx on [rmin(a;rmin(b;c)), b] - ∫ f[x] dx on [rmin(a;rmin(b;c)), a])
= ((∫ f[x] dx on [rmin(a;rmin(b;c)), c] - ∫ f[x] dx on [rmin(a;rmin(b;c)), a])
  + (∫ f[x] dx on [rmin(a;rmin(b;c)), b] - ∫ f[x] dx on [rmin(a;rmin(b;c)), c]))

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Latex:

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Latex:

1.  [a]  :  \mBbbR{}
2.  [b]  :  \mBbbR{}
3.  [c]  :  \mBbbR{}
4.  rmin(a;rmin(b;c))  \mleq{}  rmax(a;rmax(b;c))
5.  [f]  :  \{f:[rmin(a;rmin(b;c)),  rmax(a;rmax(b;c))]  {}\mrightarrow{}\mBbbR{}| 
                    ifun(f;[rmin(a;rmin(b;c)),  rmax(a;rmax(b;c))])\} 
\mvdash{}  a\_\mint{}\msupminus{}b  f[x]  dx  =  (a\_\mint{}\msupminus{}c  f[x]  dx  +  c\_\mint{}\msupminus{}b  f[x]  dx)


By

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Latex:
((InstLemma  `integral-additive-lemma`        [\mkleeneopen{}rmin(a;rmin(b;c))\mkleeneclose{};\mkleeneopen{}rmax(a;rmax(b;c))\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  (RWO  "-1"  0  THENA  Auto)
  )

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