<?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="E1:46"/><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="E2:46"/>
( <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#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="46_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_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="E2:46_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="E3:46_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="E4:46_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#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="E5:46_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#K2">Seg</a> <font color="Maroon">n</font> holds <br/> <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_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><a NAME="E1:46_1_1"/>
<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>
 ;<br/>

<a NAME="E2:46_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:46_1_1"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:46_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T18">EXTREAL1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:46_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/>;<br/>
<a NAME="E7:46_1"/><b>then </b><i><font color="Green">E70</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#E1:46_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="convfun1.html#E1:46"><i><font color="Green">E34</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_2"/>
( 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">f</font> <a href="supinf_2.html#K13">.</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  or  ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <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> &amp; <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</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> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_1"><b>suppose </b></a><a NAME="E1:46_1_2_1"/><i><font color="Green">E71</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">f</font> <a href="supinf_2.html#K13">.</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:46_1_2_1"/><i><font color="Green">E72</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">f</font> <a href="supinf_2.html#K13">.</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> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_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><a NAME="E1:46_1_2_1_1"/>
<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>
 ;<br/>

<a NAME="E2:46_1_2_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:46_1_2_1_1"/>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:46_1_2_1_1"/>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E3:46_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="supinf_2.html#T2">SUPINF_2:2</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">V</font> =  <a href="convfun1.html#K3" target="_self">Prod_of_RLS</a> <font color="Maroon">X</font>,<a href="convfun1.html#K4" target="_self">RLS_Real</a>  as   <a href="rlvect_1.html#NM2">RealLinearSpace</a> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>;<br/><a NAME="E4:46_1_2_1"/><i><font color="Green">E74</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/> ex <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">v</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_1_2"><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="E1:46_1_2_1_2"/>
<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>
 ;<br/>

<b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">v</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/>
<b>take </b>
<font color="Maroon">v</font>
;<br/>

<b>thus </b><a NAME="E4:46_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">v</font>]
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">g</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:46_1_2_1"/><i><font color="Green">E75</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E7:46_1_2_1"/><i><font color="Green">E76</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">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1">POLYNOM2:sch 1</a>(<a class="txt" href="convfun1.html#E3:46_1"><i><font color="Green">E45</font></i></a>);<br/></i><a NAME="E8:46_1_2_1"/><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="convfun1.html#E4:46_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> = <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">G</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E10:46_1_2_1"/><i><font color="Green">E78</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E11:46_1_2_1"/><i><font color="Green">E79</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">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i><b>reconsider </b><font color="Maroon">M</font> =  <a href="convfun1.html#K6" target="_self">epigraph</a> <font color="Maroon">f</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> ;<br/><a NAME="E13:46_1_2_1"/><i><font color="Green">E102</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">G</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">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_1_3"><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="E1:46_1_2_1_3"/><i><font color="Green">E103</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>
 ;<br/>

<b>hence </b><a NAME="E2:46_1_2_1_3"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E3:46_1_2_1_3"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">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="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, </i>;<br/>

<b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, </i>;<br/>
<a NAME="E5:46_1_2_1_3"/>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">w</font>
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E6:46_1_2_1_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">a</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Olive">a</font> </span>}</span> 
 
;<br/>
<a NAME="E7:46_1_2_1_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
;<br/>
<b>hence </b><a NAME="E8:46_1_2_1_3"/>
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="convfun1.html#E5:46_1"><i><font color="Green">E47</font></i></a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E14:46_1_2_1"/>
<font color="Maroon">M</font> is <a href="convex1.html#V1">convex</a>
 <b>by </b><i>, </i>;<br/><a NAME="E15:46_1_2_1"/><b>then </b><i><font color="Green">E105</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="convfun1.html#E1:46_1"><i><font color="Green">E38</font></i></a>, , , <a class="ref" href="convfun1.html#D2" target="_self">Def2</a>, , </i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><a NAME="E16:46_1_2_1"/><i><font color="Green">E106</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/> ex <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Olive">w</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_1_4"><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="E1:46_1_2_1_4"/><i><font color="Green">E107</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>
 ;<br/>

<a NAME="E2:46_1_2_1_4"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">F</font> <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"><font color="Maroon">p</font> <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"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, </i>;<br/>
<a NAME="E4:46_1_2_1_4"/>
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">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="Maroon">i</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a>  ;<br/>
<b>take </b>
<font color="Maroon">w</font>
;<br/>

<b>thus </b><a NAME="E6:46_1_2_1_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">w</font>]
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">F1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a> <b> such that </b><br/><a NAME="E18:46_1_2_1"/><i><font color="Green">E109</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E19:46_1_2_1"/><i><font color="Green">E110</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">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1">POLYNOM2:sch 1</a>();<br/></i><a NAME="E20:46_1_2_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 <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">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_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="E1:46_1_2_1_5"/><i><font color="Green">E127</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>
 ;<br/>

<a NAME="E2:46_1_2_1_5"/><b>then </b>
<font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</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/>
<b>hence </b><a NAME="E3:46_1_2_1_5"/>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</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>, </i>;<br/>


</div><b>end;</b></div><a NAME="E21:46_1_2_1"/><i><font color="Green">E128</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</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="E22:46_1_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">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_1_6"><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="E1:46_1_2_1_6"/><i><font color="Green">E129</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>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E3:46_1_2_1_6"/>
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E5:46_1"><i><font color="Green">E47</font></i></a>, </i>;<br/>
<a NAME="E4:46_1_2_1_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <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"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:46_1_2_1_6"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:46_1_2_1_6"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:46_1_2_1_6"/>
<font color="Maroon">F</font> <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"><font color="Maroon">p</font> <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"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:46_1_2_1_6"/><b>then </b>
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">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="Maroon">i</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:46_1_2_1_6"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E23:46_1_2_1"/><b>then </b>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F1</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="convfun1.html#E1:46_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="convfun1.html#D2" target="_self">Def2</a>, , </i>;<br/><a NAME="E24:46_1_2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="convfun1.html#E1:46_1"><i><font color="Green">E38</font></i></a>, , , , </i>;<br/><a NAME="E25:46_1_2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">w</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Olive">w</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">w</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E27:46_1_2_1"/><i><font color="Green">E133</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E28:46_1_2_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">w</font>
 ;<br/><a NAME="E29:46_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">z</font> &amp; <font color="Maroon">w</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="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E30:46_1_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>, <a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_2"><b>suppose </b></a><a NAME="E1:46_1_2_2"/>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <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> &amp; <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</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> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</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="E3:46_1_2_2"/><i><font color="Green">E135</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>and </b><br/><a NAME="E4:46_1_2_2"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><a NAME="E5:46_1_2_2"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">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="Maroon">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="Maroon">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="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> <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="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"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E6:46_1_2_2"/><i><font color="Green">E143</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">F</font>
 <b>by </b><i><a class="txt" href="convfun1.html#E1:46_1"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:46_1_2_2"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">p</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#E2:46_1"><i><font color="Green">E41</font></i></a>, </i>;<br/><a NAME="E8:46_1_2_2"/><b>then </b>
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="txt" href="convfun1.html#E3:46_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/><a NAME="E9:46_1_2_2"/><b>then </b>
 <a href="supinf_1.html#K1">+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#E4:46_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E10:46_1_2_2"/><b>then </b>
 <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E11:46_1_2_2"/>
<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="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E3:46"/>
( ( 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> ) 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="46_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_2"/><i><font color="Green">E144</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#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>
 ;<br/>

<a NAME="E2:46_2"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><br/> for <font color="Olive">q</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;</a> 1 holds <br/><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">q</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">q</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Olive">q</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">q</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">x2</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Maroon">x2</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:46_2_1"/><i><font color="Green">E145</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">q</font>,<span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">q</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> 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> ;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">q</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> 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> ;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">q</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>  ;<br/>
<b>set </b><font color="Maroon">n</font> = 2;<br/>
<a NAME="E6:46_2_1"/><i><font color="Green">E146</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 2 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 2 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<i><font color="Green">E151</font></i>:  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">p</font> = 
<font color="Maroon">q</font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T107">RVSUM_1:107</a></i>
<br/>.= 
1

;<br/>
<a NAME="E8:46_2_1"/><i><font color="Green">E152</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> 2 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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2_1_2"><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="E1:46_2_1_2"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_2_1_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:46_2_1_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:46_2_1_2_1_1"/><b>then </b>
( <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">q</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">x1</font> &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">q</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">F</font> <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> <font color="Maroon">q</font></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> <font color="Maroon">x1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a>, <a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/><b>hence </b><a NAME="E3:46_2_1_2_1_1"/>
( <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">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="Maroon">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="Maroon">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="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> <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="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"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:46_2_1_2_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:46_2_1_2_1_2"/><b>then </b>
( <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K10">-</a> <font color="Maroon">q</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">x2</font> &amp; <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">q</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">F</font> <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">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">q</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> <font color="Maroon">x2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a>, <a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/><b>hence </b><a NAME="E3:46_2_1_2_1_2"/>
( <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">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="Maroon">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="Maroon">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="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> <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="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"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E9:46_2_1"/>
(  <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"><font color="Maroon">q</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">q</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span> &amp;  <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"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">q</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T62">RLVECT_1:62</a></i>;<br/>
<b>hence </b><a NAME="E10:46_2_1"/>
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">q</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">q</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:46_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>
