<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/>



<b>assume </b><a NAME="E1:48"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>  holds <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/>

<b>thus </b><a NAME="E2:48"/>
( <font color="Maroon">f</font> is <a href="convfun1.html#V1" target="_self">convex</a> implies for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> <br/> for <font color="Olive">F</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <br/> for <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp;  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">n</font> holds <br/>( <font color="Olive">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">F</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Olive">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span></span>)</span> ) ) holds <br/><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">z</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Olive">F</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> is <a href="convfun1.html#V1" target="_self">convex</a>
 ;<br/>

<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> ;<br/><b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>, <font color="Maroon">z</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E2:48_1"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>and </b><br/><a NAME="E3:48_1"/><i><font color="Green">E45</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E4:48_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/>( <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span></span>)</span> )
 ;<br/>

<b>consider </b><font color="Maroon">s</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>, <font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:48_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> holds <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> iff <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i/>;<br/>
<a NAME="E7:48_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E8:48_1"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:48_1"/><i><font color="Green">E70</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:48_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">s</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <b>by </b><i>, , <a class="ref" href="finseq_2.html#T28">FINSEQ_2:28</a></i>;<br/>
<a NAME="E12:48_1"/><i><font color="Green">E72</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E3:48_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:48_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">p</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>, <a href="supinf_1.html#K3">REAL</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T30">FINSEQ_2:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">s1</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T36">FINSEQ_2:36</a></i>;<br/>
<a NAME="E15:48_1"/><i><font color="Green">E74</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E4:48_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="convfun1.html#E6:48_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_2.html#T37">FINSEQ_2:37</a></i>;<br/>
<a NAME="E16:48_1"/><i><font color="Green">E75</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E17:48_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">z</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T30">FINSEQ_2:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">z</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">s1</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="finseq_2.html#T36">FINSEQ_2:36</a></i>;<br/>
<a NAME="E19:48_1"/><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E4:48_1"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="finseq_2.html#T37">FINSEQ_2:37</a></i>;<br/>
<a NAME="E20:48_1"/><i><font color="Green">E78</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">z</font>
 
<b>by </b><i>, <a class="ref" href="convfun1.html#D2" target="_self">Def2</a>, <a class="ref" href="rlvect_2.html#T9">RLVECT_2:9</a></i>;<br/>
<a NAME="E21:48_1"/><i><font color="Green">E79</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E22:48_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">y</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T30">FINSEQ_2:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">s1</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="finseq_2.html#T36">FINSEQ_2:36</a></i>;<br/>
<a NAME="E24:48_1"/><i><font color="Green">E103</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E4:48_1"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="finseq_2.html#T37">FINSEQ_2:37</a></i>;<br/>
<a NAME="E25:48_1"/><i><font color="Green">E105</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E26:48_1"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">F</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>, <a href="numbers.html#K6">ExtREAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T30">FINSEQ_2:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">s1</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T36">FINSEQ_2:36</a></i>;<br/>
<a NAME="E28:48_1"/><i><font color="Green">E107</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E4:48_1"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="finseq_2.html#T37">FINSEQ_2:37</a></i>;<br/>
<a NAME="E29:48_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/>( <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:48_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E30:48_1"/><b>then </b><i><font color="Green">E108</font></i>: 
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , <a class="txt" href="convfun1.html#E1:48"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E31:48_1"/><b>then </b><i><font color="Green">E109</font></i>: 
 <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , <a class="ref" href="convfun1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E32:48_1"/>
<font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T21">RFINSEQ:21</a></i>;<br/>
<a NAME="E33:48_1"/><b>then </b><i><font color="Green">E110</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>;<br/>
<a NAME="E34:48_1"/><i><font color="Green">E126</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font> holds <br/>( <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:48_1_1"/><i><font color="Green">E127</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E2:48_1_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:48_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E4:48_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T19">INT_1:19</a></i>;<br/>
<a NAME="E5:48_1_1"/><b>then </b><i><font color="Green">E132</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="convfun1.html#T2" target="_self">Th2</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E6:48_1_1"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:48_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="convfun1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E8:48_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:48_1_1"/><b>then </b>
not <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E10:48_1_1"/>
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, , <a class="txt" href="convfun1.html#E6"><i><font color="Green">Lemma66</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:48_1"/><i><font color="Green">E134</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_2_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_1_1"><b>suppose </b></a><a NAME="E1:48_1_2_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_2_1_1"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="convfun1.html#E8:48_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_5.html#T35">FINSEQ_5:35</a>, <a class="ref" href="rvsum_1.html#T102">RVSUM_1:102</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_1_2"><b>suppose </b></a><a NAME="E1:48_1_2_1_2"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:48_1_2_1_2"/>
for <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> holds <br/><font color="Olive">w</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_2_1_2_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_1_2_1_2_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span>
 <b>and </b><br/><a NAME="E4:48_1_2_1_2_1"/><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<a NAME="E7:48_1_2_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E8:48_1"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E8:48_1_2_1_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:48_1_2_1_2_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:48_1_2_1_2_1"/><b>then </b><i><font color="Green">E143</font></i>: 
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E11:48_1_2_1_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p'</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E8:48_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:48_1_2_1_2_1"/><b>then </b>
<font color="Maroon">p'</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E13:48_1_2_1_2_1"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="convfun1.html#E8:48_1"><i><font color="Green">E48</font></i></a>, , , <a class="txt" href="convfun1.html#E6"><i><font color="Green">Lemma66</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<b>hence </b><a NAME="E14:48_1_2_1_2_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:48_1_2_1_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E4:48_1_2_1_2"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E36:48_1"/>
<font color="Maroon">p'</font>,<font color="Maroon">p</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="rfinseq.html#T17">RFINSEQ:17</a></i>;<br/>
<a NAME="E37:48_1"/><b>then </b><i><font color="Green">E144</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , <a class="ref" href="convfun1.html#T2" target="_self">Th2</a>, <a class="ref" href="rfinseq.html#T22">RFINSEQ:22</a></i>;<br/>
<a NAME="E38:48_1"/>
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T21">RFINSEQ:21</a></i>;<br/>
<a NAME="E39:48_1"/><b>then </b><i><font color="Green">E145</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">z'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">z'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>;<br/>
<a NAME="E40:48_1"/>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_3_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_3_1_1"><b>suppose </b></a><a NAME="E1:48_1_3_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:48_1_3_1_1"/><b>then </b>
<font color="Maroon">z'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="finseq_5.html#T35">FINSEQ_5:35</a></i>;<br/><b>hence </b><a NAME="E3:48_1_3_1_1"/>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T60">RLVECT_1:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_3_1_2"><b>suppose </b></a><a NAME="E1:48_1_3_1_2"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:48_1_3_1_2"/>
for <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> holds <br/><font color="Olive">w</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_3_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_3_1_2_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_1_3_1_2_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span>
 <b>and </b><br/><a NAME="E4:48_1_3_1_2_1"/><i><font color="Green">E152</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="convfun1.html#E6"><i><font color="Green">Lemma66</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<a NAME="E7:48_1_3_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font>
 
<b>by </b><i>, <a class="txt" href="convfun1.html#E6"><i><font color="Green">Lemma66</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E8:48_1_3_1_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:48_1_3_1_2_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:48_1_3_1_2_1"/><b>then </b><i><font color="Green">E153</font></i>: 
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E11:48_1_3_1_2_1"/><b>then </b>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E12:48_1_3_1_2_1"/><b>then </b>
<font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:48_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E13:48_1_3_1_2_1"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>;<br/>
<a NAME="E14:48_1_3_1_2_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z'</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E15:48_1_3_1_2_1"/><b>then </b>
<font color="Maroon">z'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E9:48_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E16:48_1_3_1_2_1"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="txt" href="convfun1.html#E6"><i><font color="Green">Lemma66</font></i></a>, , , <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<b>hence </b><a NAME="E17:48_1_3_1_2_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:48_1_3_1_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E4:48_1_3_1_2"/>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E41:48_1"/><b>then </b><i><font color="Green">E155</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">z</font>
 
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>;<br/>
<a NAME="E42:48_1"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">F'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T21">RFINSEQ:21</a></i>;<br/>
<a NAME="E43:48_1"/>
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F'</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T25">FUNCT_1:25</a></i>;<br/>
<a NAME="E44:48_1"/><b>then </b>
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">F'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E45:48_1"/><b>then </b>
( not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> &amp; not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E46:48_1"/><b>then </b><i><font color="Green">E157</font></i>: 
 <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">F'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E47:48_1"/>
 <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_4_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_1"><b>suppose </b></a><a NAME="E1:48_1_4_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_4_1_1"/>
 <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, , <a class="ref" href="finseq_5.html#T35">FINSEQ_5:35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_4_1_2"><b>suppose </b></a><a NAME="E1:48_1_4_1_2"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:48_1_4_1_2"/>
for <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> holds <br/><font color="Olive">w</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_2.html#K1">0.</a> </span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_4_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_4_1_2_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_1_4_1_2_1"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span>
 <b>and </b><br/><a NAME="E4:48_1_4_1_2_1"/><i><font color="Green">E160</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="convfun1.html#E9:48_1"><i><font color="Green">E70</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<a NAME="E7:48_1_4_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font>
 
<b>by </b><i>, , <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E8:48_1_4_1_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E9:48_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:48_1_4_1_2_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:48_1_4_1_2_1"/><b>then </b><i><font color="Green">E161</font></i>: 
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E11:48_1_4_1_2_1"/><b>then </b>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E12:48_1_4_1_2_1"/><b>then </b>
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:48_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="convfun1.html#E12:48_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E13:48_1_4_1_2_1"/><b>then </b>
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="supinf_2.html#K1">0.</a>  <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/>
<a NAME="E14:48_1_4_1_2_1"/><b>then </b><i><font color="Green">E162</font></i>: 
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i><a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/>
<a NAME="E15:48_1_4_1_2_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F'</font>
 
<b>by </b><i>, <a class="txt" href="convfun1.html#E12:48_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E16:48_1_4_1_2_1"/><b>then </b>
<font color="Maroon">F'</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i><a class="txt" href="convfun1.html#E13:48_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E17:48_1_4_1_2_1"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i>, , <a class="txt" href="convfun1.html#E9:48_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="convfun1.html#E10:48_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<b>hence </b><a NAME="E18:48_1_4_1_2_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_2.html#K1">0.</a> </span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:48_1_4_1_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_2.html#K1">0.</a> </span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E4:48_1_4_1_2"/>
 <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E48:48_1"/><b>then </b><i><font color="Green">E168</font></i>: 
 <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="convfun1.html#T1" target="_self">Th1</a>, , <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/>
<b>set </b><font color="Maroon">k'</font> =  <a href="rfunct_3.html#K1">min</a> <font color="Maroon">k</font>,<font color="Maroon">n</font>;<br/>
<b>reconsider </b><font color="Maroon">k'</font> =  <a href="rfunct_3.html#K1">min</a> <font color="Maroon">k</font>,<font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E50:48_1"/><i><font color="Green">E169</font></i>: 
( <font color="Maroon">p'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="rfunct_3.html#K23">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">y'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">z'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">F'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E51:48_1"/><b>then </b><i><font color="Green">E170</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k'</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k'</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k'</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k'</font> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E8:48_1"><i><font color="Green">E48</font></i></a>, , , , <a class="ref" href="finseq_2.html#T24">FINSEQ_2:24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k'</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a>, <a class="ref" href="rvsum_1.html#T102">RVSUM_1:102</a></i>;<br/>
<a NAME="E53:48_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k'</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_5"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k'</font>
 ;<br/>

<a NAME="E2:48_1_5"/><b>then </b><i><font color="Green">E172</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E10:48_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:48_1_5"/>
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">p'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">z'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">F'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E9:48_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:48_1_5"/><b>then </b><i><font color="Green">E198</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z'</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F'</font> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E13:48_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:48_1_5"/><b>then </b><i><font color="Green">E199</font></i>: 
( <font color="Maroon">p'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">y'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">z'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E6:48_1_5"/><i><font color="Green">E200</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E13:48_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/>
<a NAME="E7:48_1_5"/><i><font color="Green">E201</font></i>: 
( <font color="Maroon">p'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">y'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">z'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">F'</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E15:48_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E8:48_1_5"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E8:48_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="convfun1.html#E15:48_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:48_1_5"/><b>then </b><i><font color="Green">E203</font></i>: 
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E10:48_1_5"/><b>then </b><i><font color="Green">E204</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> )
 
<b>by </b><i>, <a class="txt" href="convfun1.html#E13:48_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="convfun1.html#E16:48_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="convfun1.html#E17:48_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="convfun1.html#E19:48_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E11:48_1_5"/>
<font color="Maroon">k'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="square_1.html#T35">SQUARE_1:35</a></i>;<br/>
<a NAME="E12:48_1_5"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k'</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E13:48_1_5"/><b>then </b>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, , <a class="txt" href="convfun1.html#E12:48_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="convfun1.html#E20:48_1"><i><font color="Green">E78</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:48_1_5"/>
( <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="convfun1.html#E2:48_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="convfun1.html#E21:48_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="convfun1.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E54:48_1"/>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="txt" href="convfun1.html#E1:48_1"><i><font color="Green">E38</font></i></a>, , <a class="txt" href="convfun1.html#E6"><i><font color="Green">Lemma66</font></i></a>, , <a class="txt" href="convfun1.html#E10:48_1"><i><font color="Green">E71</font></i></a>, </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:48"/>
( ( for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> <br/> for <font color="Olive">F</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <br/> for <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp;  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">n</font> holds <br/>( <font color="Olive">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">F</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Olive">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span></span>)</span> ) ) holds <br/><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">z</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Olive">F</font> ) implies <font color="Maroon">f</font> is <a href="convfun1.html#V1" target="_self">convex</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_2"/><i><font color="Green">E205</font></i>: 
for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> <br/> for <font color="Olive">F</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <br/> for <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp;  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">n</font> holds <br/>( <font color="Olive">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">F</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Olive">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span></span>)</span> ) ) holds <br/><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">z</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Olive">F</font>
 ;<br/>

<a NAME="E2:48_2"/>
for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> <br/> for <font color="Olive">F</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <br/> for <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp;  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">n</font> holds <br/>( <font color="Olive">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">F</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Olive">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span></span>)</span> ) ) holds <br/><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">z</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Olive">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> ;<br/><b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>, <font color="Maroon">z</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:48_2_1"/><i><font color="Green">E206</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 )
 <b>and </b><br/><a NAME="E2:48_2_1"/><i><font color="Green">E207</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/>( <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E3:48_2_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/>( <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:48_2_1"/>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:48_2"/>
<font color="Maroon">f</font> is <a href="convfun1.html#V1" target="_self">convex</a>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>


</div>
