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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>



<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E36">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font>  holds <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 ;<br/>

<b>thus </b><a NAME="E2:48"/>
( <font color="Maroon" title="c2">f</font> is <a href="convfun1.html#V1" title="CONVFUN1:attr.1">convex</a> implies for <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> <br/> for <font color="Olive" title="b3">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <br/> for <font color="Olive" title="b4">y</font>, <font color="Olive" title="b5">z</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp;  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; ( for <font color="Olive" title="b6">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">n</font> holds <br/>( <font color="Olive" title="b2">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b6">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Olive" title="b5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b6">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b6">i</font></span>)</span> &amp; <font color="Olive" title="b3">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b6">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b6">i</font></span>)</span></span>)</span> ) ) holds <br/><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Olive" title="b5">z</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Olive" title="b3">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" title="E37">A2</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="convfun1.html#V1" title="CONVFUN1:attr.1">convex</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c4">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c5">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">z</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E2:48_1"/><i><font color="Green" title="E38">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> )
 <b>and </b><br/><a NAME="E3:48_1"/><i><font color="Green" title="E39">A4</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>and </b><br/><a NAME="E4:48_1"/><i><font color="Green" title="E40">A5</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font> holds <br/>( <font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c7">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">s</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c9">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:48_1"/><i><font color="Green" title="E41">A6</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font> holds <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">k</font> iff <font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="convfun1.html#E40"><i><font color="Green" title="E35">Lm20</font></i></a></i>;<br/>
<a NAME="E7:48_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E8:48_1"/><b>then </b><i><font color="Green" title="E42">A7</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:48_1"/><i><font color="Green" title="E43">A8</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:48_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:48_1"/><b>then </b><i><font color="Green" title="E44">A9</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E8:48_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">s1</font> = <font color="Maroon" title="c8">s</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font> <b>by </b><i><a class="txt" href="convfun1.html#E8:48_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="convfun1.html#E9:48_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="finseq_2.html#T28" title="FINSEQ_2:th.28">FINSEQ_2:28</a></i>;<br/>
<a NAME="E12:48_1"/><i><font color="Green" title="E45">A10</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E10:48_1"><i><font color="Green" title="E44">A9</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:48_1"/><i><font color="Green" title="E46">A11</font></i>: 
<font color="Maroon" title="c4">p</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>, <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 
<b>by </b><i><a class="txt" href="convfun1.html#E9:48_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="finseq_2.html#T30" title="FINSEQ_2:th.30">FINSEQ_2:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">p'</font> = <font color="Maroon" title="c4">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">s1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T36" title="FINSEQ_2:th.36">FINSEQ_2:36</a></i>;<br/>
<a NAME="E15:48_1"/><i><font color="Green" title="E47">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E12:48_1"><i><font color="Green" title="E45">A10</font></i></a>, <a class="txt" href="convfun1.html#E13:48_1"><i><font color="Green" title="E46">A11</font></i></a>, <a class="ref" href="finseq_2.html#T37" title="FINSEQ_2:th.37">FINSEQ_2:37</a></i>;<br/>
<a NAME="E16:48_1"/><i><font color="Green" title="E48">A13</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:48_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E17:48_1"/><b>then </b><i><font color="Green" title="E49">A14</font></i>: 
<font color="Maroon" title="c7">z</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T30" title="FINSEQ_2:th.30">FINSEQ_2:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">z'</font> = <font color="Maroon" title="c7">z</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">s1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="finseq_2.html#T36" title="FINSEQ_2:th.36">FINSEQ_2:36</a></i>;<br/>
<a NAME="E19:48_1"/><i><font color="Green" title="E50">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E12:48_1"><i><font color="Green" title="E45">A10</font></i></a>, <a class="txt" href="convfun1.html#E17:48_1"><i><font color="Green" title="E49">A14</font></i></a>, <a class="ref" href="finseq_2.html#T37" title="FINSEQ_2:th.37">FINSEQ_2:37</a></i>;<br/>
<a NAME="E20:48_1"/><i><font color="Green" title="E51">A16</font></i>: 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c12">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c7">z</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E9:48_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="txt" href="convfun1.html#E16:48_1"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="rlvect_2.html#T9" title="RLVECT_2:th.9">RLVECT_2:9</a></i>;<br/>
<a NAME="E21:48_1"/><i><font color="Green" title="E52">A17</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:48_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E22:48_1"/><b>then </b><i><font color="Green" title="E53">A18</font></i>: 
<font color="Maroon" title="c6">y</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T30" title="FINSEQ_2:th.30">FINSEQ_2:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">y'</font> = <font color="Maroon" title="c6">y</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">s1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="finseq_2.html#T36" title="FINSEQ_2:th.36">FINSEQ_2:36</a></i>;<br/>
<a NAME="E24:48_1"/><i><font color="Green" title="E54">A19</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E12:48_1"><i><font color="Green" title="E45">A10</font></i></a>, <a class="txt" href="convfun1.html#E22:48_1"><i><font color="Green" title="E53">A18</font></i></a>, <a class="ref" href="finseq_2.html#T37" title="FINSEQ_2:th.37">FINSEQ_2:37</a></i>;<br/>
<a NAME="E25:48_1"/><i><font color="Green" title="E55">A20</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:48_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E26:48_1"/><b>then </b><i><font color="Green" title="E56">A21</font></i>: 
<font color="Maroon" title="c5">F</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T30" title="FINSEQ_2:th.30">FINSEQ_2:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">F'</font> = <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">s1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T36" title="FINSEQ_2:th.36">FINSEQ_2:36</a></i>;<br/>
<a NAME="E28:48_1"/><i><font color="Green" title="E57">A22</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">F'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E12:48_1"><i><font color="Green" title="E45">A10</font></i></a>, <a class="txt" href="convfun1.html#E26:48_1"><i><font color="Green" title="E56">A21</font></i></a>, <a class="ref" href="finseq_2.html#T37" title="FINSEQ_2:th.37">FINSEQ_2:37</a></i>;<br/>
<a NAME="E29:48_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font> holds <br/>( <font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E4:48_1"><i><font color="Green" title="E40">A5</font></i></a></i>;<br/>
<a NAME="E30:48_1"/><b>then </b><i><font color="Green" title="E58">A23</font></i>: 
not  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:48"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="convfun1.html#E2:48_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="convfun1.html#E38"><i><font color="Green" title="E33">Lm19</font></i></a></i>;<br/>
<a NAME="E31:48_1"/><b>then </b><i><font color="Green" title="E59">A24</font></i>: 
 <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c14">F'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c5">F</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E9:48_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="txt" href="convfun1.html#E25:48_1"><i><font color="Green" title="E55">A20</font></i></a>, <a class="ref" href="convfun1.html#T8" target="_self" title="CONVFUN1:th.8">Th8</a></i>;<br/>
<a NAME="E32:48_1"/>
<font color="Maroon" title="c11">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T21" title="RFINSEQ:th.21">RFINSEQ:21</a></i>;<br/>
<a NAME="E33:48_1"/><b>then </b><i><font color="Green" title="E60">A25</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c11">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>;<br/>
<a NAME="E34:48_1"/><i><font color="Green" title="E61">A26</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c9">k</font> holds <br/>( <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c15">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:48_1_1"/><i><font color="Green" title="E62">A27</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font>
 <b>and </b><br/><a NAME="E2:48_1_1"/><i><font color="Green" title="E63">A28</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c9">k</font>
 ;<br/>

<a NAME="E3:48_1_1"/><i><font color="Green" title="E64">A29</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:48_1_1"><i><font color="Green" title="E63">A28</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E4:48_1_1"/>
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T19" title="INT_1:th.19">INT_1:19</a></i>;<br/>
<a NAME="E5:48_1_1"/><b>then </b><i><font color="Green" title="E65">A30</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:48_1_1"><i><font color="Green" title="E62">A27</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:48_1_1"/><b>then </b><i><font color="Green" title="E66">A31</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E9:48_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="txt" href="convfun1.html#E3:48_1_1"><i><font color="Green" title="E64">A29</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:48_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">i</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:48_1_1"><i><font color="Green" title="E62">A27</font></i></a></i>;<br/>
<a NAME="E8:48_1_1"/><b>then </b>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:48_1_1"/><b>then </b>
not <font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E10:48_1_1"/>
( <font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="convfun1.html#E6:48_1"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="convfun1.html#E3:48_1_1"><i><font color="Green" title="E64">A29</font></i></a>, <a class="txt" href="convfun1.html#E5:48_1_1"><i><font color="Green" title="E65">A30</font></i></a>, <a class="txt" href="convfun1.html#E6:48_1_1"><i><font color="Green" title="E66">A31</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:48_1"/><i><font color="Green" title="E62">A32</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">n</font> or <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">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" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_2_1_1"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="convfun1.html#E15:48_1"><i><font color="Green" title="E47">A12</font></i></a>, <a class="ref" href="finseq_5.html#T35" title="FINSEQ_5:th.35">FINSEQ_5:35</a>, <a class="ref" href="rvsum_1.html#T102" title="RVSUM_1:th.102">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" title="E63">A33</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">n</font>
 ;<br/><div class="add"><a NAME="E2:48_1_2_1_2"/>
for <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> holds <br/><font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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" title="c15">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_2_1_2_1"/>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:48_1_2_1_2_1"/><i><font color="Green" title="E64">A34</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span>
 <b>and </b><br/><a NAME="E4:48_1_2_1_2_1"/><i><font color="Green" title="E65">A35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">w</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c16">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="convfun1.html#E3:48_1_2_1_2_1"><i><font color="Green" title="E64">A34</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">k1</font> = <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c9">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="convfun1.html#E1:48_1_2_1_2"><i><font color="Green" title="E63">A33</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/>
<a NAME="E7:48_1_2_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">k1</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E15:48_1"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="convfun1.html#E1:48_1_2_1_2"><i><font color="Green" title="E63">A33</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E8:48_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c18">k1</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E3:48_1_2_1_2_1"><i><font color="Green" title="E64">A34</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c9">k</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:48_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E66">A36</font></i>: 
( <font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="convfun1.html#E34:48_1"><i><font color="Green" title="E61">A26</font></i></a></i>;<br/>
<a NAME="E11:48_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p'</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E15:48_1"><i><font color="Green" title="E47">A12</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:48_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c11">p'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="convfun1.html#E10:48_1_2_1_2_1"><i><font color="Green" title="E66">A36</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E13:48_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="convfun1.html#E15:48_1"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="convfun1.html#E1:48_1_2_1_2"><i><font color="Green" title="E63">A33</font></i></a>, <a class="txt" href="convfun1.html#E3:48_1_2_1_2_1"><i><font color="Green" title="E64">A34</font></i></a>, <a class="txt" href="convfun1.html#E4:48_1_2_1_2_1"><i><font color="Green" title="E65">A35</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<b>hence </b><a NAME="E14:48_1_2_1_2_1"/>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:48_1_2_1_2"/><b>then </b>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E4:48_1_2_1_2"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="convfun1.html#E28"><i><font color="Green" title="E26">Lm15</font></i></a></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" title="c11">p'</font>,<font color="Maroon" title="c4">p</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="rfinseq.html#T17" title="RFINSEQ:th.17">RFINSEQ:17</a></i>;<br/>
<a NAME="E37:48_1"/><b>then </b><i><font color="Green" title="E63">A37</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="convfun1.html#E3:48_1"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="convfun1.html#E33:48_1"><i><font color="Green" title="E60">A25</font></i></a>, <a class="txt" href="convfun1.html#E35:48_1"><i><font color="Green" title="E62">A32</font></i></a>, <a class="ref" href="rfinseq.html#T22" title="RFINSEQ:th.22">RFINSEQ:22</a></i>;<br/>
<a NAME="E38:48_1"/>
<font color="Maroon" title="c12">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T21" title="RFINSEQ:th.21">RFINSEQ:21</a></i>;<br/>
<a NAME="E39:48_1"/><b>then </b><i><font color="Green" title="E64">A38</font></i>: 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c12">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></i>;<br/>
<a NAME="E40:48_1"/>
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">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" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">n</font> or <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">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" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">n</font>
 ;<br/><div class="add"><a NAME="E2:48_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c12">z'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="convfun1.html#E19:48_1"><i><font color="Green" title="E50">A15</font></i></a>, <a class="ref" href="finseq_5.html#T35" title="FINSEQ_5:th.35">FINSEQ_5:35</a></i>;<br/><b>hence </b><a NAME="E3:48_1_3_1_1"/>
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T60" title="RLVECT_1:th.60">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" title="E65">A39</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">n</font>
 ;<br/><div class="add"><a NAME="E2:48_1_3_1_2"/>
for <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> holds <br/><font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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" title="c15">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_3_1_2_1"/>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:48_1_3_1_2_1"/><i><font color="Green" title="E66">A40</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span>
 <b>and </b><br/><a NAME="E4:48_1_3_1_2_1"/><i><font color="Green" title="E67">A41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">w</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c16">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="convfun1.html#E3:48_1_3_1_2_1"><i><font color="Green" title="E66">A40</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">k1</font> = <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c9">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="convfun1.html#E1:48_1_3_1_2"><i><font color="Green" title="E65">A39</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/>
<a NAME="E7:48_1_3_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">k1</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E19:48_1"><i><font color="Green" title="E50">A15</font></i></a>, <a class="txt" href="convfun1.html#E1:48_1_3_1_2"><i><font color="Green" title="E65">A39</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E8:48_1_3_1_2_1"/><b>then </b>
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c18">k1</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E3:48_1_3_1_2_1"><i><font color="Green" title="E66">A40</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c9">k</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:48_1_3_1_2_1"/><b>then </b><i><font color="Green" title="E68">A42</font></i>: 
( <font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="convfun1.html#E34:48_1"><i><font color="Green" title="E61">A26</font></i></a></i>;<br/>
<a NAME="E11:48_1_3_1_2_1"/><b>then </b>
<font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E9:48_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E12:48_1_3_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E4:48_1"><i><font color="Green" title="E40">A5</font></i></a></i>;<br/>
<a NAME="E13:48_1_3_1_2_1"/><b>then </b><i><font color="Green" title="E69">A43</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E10:48_1_3_1_2_1"><i><font color="Green" title="E68">A42</font></i></a>, <a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>;<br/>
<a NAME="E14:48_1_3_1_2_1"/>
<font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">z'</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E19:48_1"><i><font color="Green" title="E50">A15</font></i></a>, <a class="txt" href="convfun1.html#E10:48_1_3_1_2_1"><i><font color="Green" title="E68">A42</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E15:48_1_3_1_2_1"/><b>then </b>
<font color="Maroon" title="c12">z'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E13:48_1_3_1_2_1"><i><font color="Green" title="E69">A43</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E16:48_1_3_1_2_1"/><b>then </b>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E19:48_1"><i><font color="Green" title="E50">A15</font></i></a>, <a class="txt" href="convfun1.html#E1:48_1_3_1_2"><i><font color="Green" title="E65">A39</font></i></a>, <a class="txt" href="convfun1.html#E3:48_1_3_1_2_1"><i><font color="Green" title="E66">A40</font></i></a>, <a class="txt" href="convfun1.html#E4:48_1_3_1_2_1"><i><font color="Green" title="E67">A41</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<b>hence </b><a NAME="E17:48_1_3_1_2_1"/>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">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" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E4:48_1_3_1_2"/>
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="convfun1.html#E29"><i><font color="Green" title="E27">Lm16</font></i></a></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" title="E65">A44</font></i>: 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c7">z</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E20:48_1"><i><font color="Green" title="E51">A16</font></i></a>, <a class="txt" href="convfun1.html#E39:48_1"><i><font color="Green" title="E64">A38</font></i></a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>;<br/>
<a NAME="E42:48_1"/><i><font color="Green" title="E66">A45</font></i>: 
<font color="Maroon" title="c14">F'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T21" title="RFINSEQ:th.21">RFINSEQ:21</a></i>;<br/>
<a NAME="E43:48_1"/>
not  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">F'</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E30:48_1"><i><font color="Green" title="E58">A23</font></i></a>, <a class="ref" href="funct_1.html#T25" title="FUNCT_1:th.25">FUNCT_1:25</a></i>;<br/>
<a NAME="E44:48_1"/><b>then </b>
not  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E42:48_1"><i><font color="Green" title="E66">A45</font></i></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E45:48_1"/><b>then </b>
( not  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> &amp; not  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E46:48_1"/><b>then </b><i><font color="Green" title="E67">A46</font></i>: 
 <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c14">F'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E42:48_1"><i><font color="Green" title="E66">A45</font></i></a>, <a class="ref" href="convfun1.html#T7" target="_self" title="CONVFUN1:th.7">Th7</a></i>;<br/>
<a NAME="E47:48_1"/>
 <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">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" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">n</font> or <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">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" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_4_1_1"/>
 <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E28:48_1"><i><font color="Green" title="E57">A22</font></i></a>, <a class="ref" href="convfun1.html#T4" target="_self" title="CONVFUN1:th.4">Th4</a>, <a class="ref" href="finseq_5.html#T35" title="FINSEQ_5:th.35">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" title="E68">A47</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">n</font>
 ;<br/><div class="add"><a NAME="E2:48_1_4_1_2"/>
for <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> holds <br/><font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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" title="c15">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_4_1_2_1"/>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:48_1_4_1_2_1"/><i><font color="Green" title="E69">A48</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span>
 <b>and </b><br/><a NAME="E4:48_1_4_1_2_1"/><i><font color="Green" title="E70">A49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">w</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c16">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="convfun1.html#E3:48_1_4_1_2_1"><i><font color="Green" title="E69">A48</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">k1</font> = <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c9">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="convfun1.html#E1:48_1_4_1_2"><i><font color="Green" title="E68">A47</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/>
<a NAME="E7:48_1_4_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">k1</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E28:48_1"><i><font color="Green" title="E57">A22</font></i></a>, <a class="txt" href="convfun1.html#E1:48_1_4_1_2"><i><font color="Green" title="E68">A47</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E8:48_1_4_1_2_1"/><b>then </b>
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c18">k1</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E3:48_1_4_1_2_1"><i><font color="Green" title="E69">A48</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c9">k</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:48_1_4_1_2_1"/><b>then </b><i><font color="Green" title="E71">A50</font></i>: 
( <font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="convfun1.html#E34:48_1"><i><font color="Green" title="E61">A26</font></i></a></i>;<br/>
<a NAME="E11:48_1_4_1_2_1"/><b>then </b>
<font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E9:48_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E12:48_1_4_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 0</span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E4:48_1"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="convfun1.html#E10:48_1_4_1_2_1"><i><font color="Green" title="E71">A50</font></i></a></i>;<br/>
<a NAME="E13:48_1_4_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="measure6.html#D1" title="MEASURE6:def.1">MEASURE6:def 1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/>
<a NAME="E14:48_1_4_1_2_1"/><b>then </b><i><font color="Green" title="E72">A51</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="ref" href="extreal1.html#D1" title="EXTREAL1:def.1">EXTREAL1:def 1</a></i>;<br/>
<a NAME="E15:48_1_4_1_2_1"/>
<font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">F'</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E28:48_1"><i><font color="Green" title="E57">A22</font></i></a>, <a class="txt" href="convfun1.html#E10:48_1_4_1_2_1"><i><font color="Green" title="E71">A50</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E16:48_1_4_1_2_1"/><b>then </b>
<font color="Maroon" title="c14">F'</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="txt" href="convfun1.html#E14:48_1_4_1_2_1"><i><font color="Green" title="E72">A51</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E17:48_1_4_1_2_1"/><b>then </b>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="txt" href="convfun1.html#E28:48_1"><i><font color="Green" title="E57">A22</font></i></a>, <a class="txt" href="convfun1.html#E1:48_1_4_1_2"><i><font color="Green" title="E68">A47</font></i></a>, <a class="txt" href="convfun1.html#E3:48_1_4_1_2_1"><i><font color="Green" title="E69">A48</font></i></a>, <a class="txt" href="convfun1.html#E4:48_1_4_1_2_1"><i><font color="Green" title="E70">A49</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<b>hence </b><a NAME="E18:48_1_4_1_2_1"/>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">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" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E4:48_1_4_1_2"/>
 <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E27"><i><font color="Green" title="E25">Lm14</font></i></a></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" title="E68">A52</font></i>: 
 <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c5">F</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E31:48_1"><i><font color="Green" title="E59">A24</font></i></a>, <a class="txt" href="convfun1.html#E46:48_1"><i><font color="Green" title="E67">A46</font></i></a>, <a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">k'</font> =  <a href="rfunct_3.html#K1" title="RFUNCT_3:func.1">min</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c3">n</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">k'</font> =  <a href="rfunct_3.html#K1" title="RFUNCT_3:func.1">min</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c3">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E50:48_1"/><i><font color="Green" title="E69">A53</font></i>: 
( <font color="Maroon" title="c11">p'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p'</font> <a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">k</font></span>)</span> &amp; <font color="Maroon" title="c13">y'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">k</font></span>)</span> &amp; <font color="Maroon" title="c12">z'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">k</font></span>)</span> &amp; <font color="Maroon" title="c14">F'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">k</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E51:48_1"/><b>then </b><i><font color="Green" title="E70">A54</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">k'</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">y'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">k'</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">k'</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">k'</font> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E15:48_1"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="convfun1.html#E19:48_1"><i><font color="Green" title="E50">A15</font></i></a>, <a class="txt" href="convfun1.html#E24:48_1"><i><font color="Green" title="E54">A19</font></i></a>, <a class="txt" href="convfun1.html#E28:48_1"><i><font color="Green" title="E57">A22</font></i></a>, <a class="ref" href="finseq_2.html#T24" title="FINSEQ_2:th.24">FINSEQ_2:24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">k'</font> = <font color="Maroon" title="c16">k'</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="convfun1.html#E37:48_1"><i><font color="Green" title="E63">A37</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a>, <a class="ref" href="rvsum_1.html#T102" title="RVSUM_1:th.102">RVSUM_1:102</a></i>;<br/>
<a NAME="E53:48_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c17">k'</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">y'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">y'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">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" title="c18">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_5"/><i><font color="Green" title="E71">A55</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c17">k'</font>
 ;<br/>

<a NAME="E2:48_1_5"/><b>then </b><i><font color="Green" title="E72">A56</font></i>: 
( <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> &amp; <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">y'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> &amp; <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> &amp; <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E51:48_1"><i><font color="Green" title="E70">A54</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:48_1_5"/>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">k</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">y'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">k</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">z'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">k</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">F'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">k</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E50:48_1"><i><font color="Green" title="E69">A53</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:48_1_5"/><b>then </b><i><font color="Green" title="E73">A57</font></i>: 
( <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p'</font> &amp; <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">y'</font> &amp; <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">z'</font> &amp; <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">F'</font> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:48_1_5"><i><font color="Green" title="E72">A56</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:48_1_5"/><b>then </b><i><font color="Green" title="E74">A58</font></i>: 
( <font color="Maroon" title="c11">p'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c13">y'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c12">z'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c14">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F'</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c18">i</font> )
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E6:48_1_5"/><i><font color="Green" title="E75">A59</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c13">y'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:48_1_5"><i><font color="Green" title="E72">A56</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/>
<a NAME="E7:48_1_5"/><i><font color="Green" title="E76">A60</font></i>: 
( <font color="Maroon" title="c11">p'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">i</font></span>)</span> &amp; <font color="Maroon" title="c13">y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">i</font></span>)</span> &amp; <font color="Maroon" title="c12">z'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">i</font></span>)</span> &amp; <font color="Maroon" title="c14">F'</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">i</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E4:48_1_5"><i><font color="Green" title="E73">A57</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E8:48_1_5"/><i><font color="Green" title="E77">A61</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E15:48_1"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="convfun1.html#E4:48_1_5"><i><font color="Green" title="E73">A57</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:48_1_5"/><b>then </b><i><font color="Green" title="E78">A62</font></i>: 
<font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E9:48_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E10:48_1_5"/><b>then </b><i><font color="Green" title="E79">A63</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c13">y'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">i</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E21:48_1"><i><font color="Green" title="E52">A17</font></i></a>, <a class="txt" href="convfun1.html#E2:48_1_5"><i><font color="Green" title="E72">A56</font></i></a>, <a class="txt" href="convfun1.html#E5:48_1_5"><i><font color="Green" title="E74">A58</font></i></a>, <a class="txt" href="convfun1.html#E6:48_1_5"><i><font color="Green" title="E75">A59</font></i></a>, <a class="txt" href="convfun1.html#E7:48_1_5"><i><font color="Green" title="E76">A60</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E11:48_1_5"/>
<font color="Maroon" title="c17">k'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E12:48_1_5"/><b>then </b>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c17">k'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E13:48_1_5"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="convfun1.html#E6:48_1"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="convfun1.html#E9:48_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="txt" href="convfun1.html#E1:48_1_5"><i><font color="Green" title="E71">A55</font></i></a>, <a class="txt" href="convfun1.html#E8:48_1_5"><i><font color="Green" title="E77">A61</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:48_1_5"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">z'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">y'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">F'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">y'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="convfun1.html#E4:48_1"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="convfun1.html#E9:48_1_5"><i><font color="Green" title="E78">A62</font></i></a>, <a class="txt" href="convfun1.html#E10:48_1_5"><i><font color="Green" title="E79">A63</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E54:48_1"/>
<font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c7">z</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="convfun1.html#E1:48"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="convfun1.html#E1:48_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="convfun1.html#E37:48_1"><i><font color="Green" title="E63">A37</font></i></a>, <a class="txt" href="convfun1.html#E41:48_1"><i><font color="Green" title="E65">A44</font></i></a>, <a class="txt" href="convfun1.html#E48:48_1"><i><font color="Green" title="E68">A52</font></i></a>, <a class="txt" href="convfun1.html#E51:48_1"><i><font color="Green" title="E70">A54</font></i></a>, <a class="ref" href="convfun1.html#T13" target="_self" title="CONVFUN1:th.13">Th13</a></i>;<br/>


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

<b>thus </b><a NAME="E3:48"/>
( ( for <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> <br/> for <font color="Olive" title="b3">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <br/> for <font color="Olive" title="b4">y</font>, <font color="Olive" title="b5">z</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp;  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; ( for <font color="Olive" title="b6">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">n</font> holds <br/>( <font color="Olive" title="b2">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b6">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Olive" title="b5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b6">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b6">i</font></span>)</span> &amp; <font color="Olive" title="b3">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b6">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b6">i</font></span>)</span></span>)</span> ) ) holds <br/><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Olive" title="b5">z</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Olive" title="b3">F</font> ) implies <font color="Maroon" title="c2">f</font> is <a href="convfun1.html#V1" title="CONVFUN1:attr.1">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" title="E37">A64</font></i>: 
for <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> <br/> for <font color="Olive" title="b3">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <br/> for <font color="Olive" title="b4">y</font>, <font color="Olive" title="b5">z</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp;  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; ( for <font color="Olive" title="b6">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">n</font> holds <br/>( <font color="Olive" title="b2">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b6">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Olive" title="b5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b6">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b6">i</font></span>)</span> &amp; <font color="Olive" title="b3">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b6">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b6">i</font></span>)</span></span>)</span> ) ) holds <br/><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Olive" title="b5">z</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Olive" title="b3">F</font>
 ;<br/>

<a NAME="E2:48_2"/>
for <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> <br/> for <font color="Olive" title="b3">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <br/> for <font color="Olive" title="b4">y</font>, <font color="Olive" title="b5">z</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp;  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; ( for <font color="Olive" title="b6">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">n</font> holds <br/>( <font color="Olive" title="b2">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b6">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Olive" title="b5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b6">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b6">i</font></span>)</span> &amp; <font color="Olive" title="b3">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b6">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b6">i</font></span>)</span></span>)</span> ) ) holds <br/><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Olive" title="b5">z</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Olive" title="b3">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" title="c3">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c4">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c5">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">z</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:48_2_1"/><i><font color="Green" title="E38">A65</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> &amp;  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>and </b><br/><a NAME="E2:48_2_1"/><i><font color="Green" title="E39">A66</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font> holds <br/>( <font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c7">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E3:48_2_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font> holds <br/>( <font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c7">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:48_2_1"><i><font color="Green" title="E39">A66</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:48_2_1"/>
<font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c7">z</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="convfun1.html#E1:48_2"><i><font color="Green" title="E37">A64</font></i></a>, <a class="txt" href="convfun1.html#E1:48_2_1"><i><font color="Green" title="E38">A65</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:48_2"/>
<font color="Maroon" title="c2">f</font> is <a href="convfun1.html#V1" title="CONVFUN1:attr.1">convex</a>
 <b>by </b><i><a class="txt" href="convfun1.html#E1:48"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="convfun1.html#T13" target="_self" title="CONVFUN1:th.13">Th13</a></i>;<br/>


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


</div>
