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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">F</font>,<font color="Maroon">a</font> <a href="mesfunc3.html#R1" target="_self">are_Re-presentation_of</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E3:19"/><i><font color="Green">E37</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span> &amp;  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">a</font> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> holds <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">ax</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E5:19"/><i><font color="Green">E38</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ax</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> holds <br/><font color="Maroon">ax</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">ax</font>
;<br/>

<a NAME="E6:19"/><i><font color="Green">E39</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">ax</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:19"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">ax</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:19_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:19_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:19_1"/>
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="mesfunc3.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E7:19_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mesfunc3.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">SA</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E9:19_1"/><i><font color="Green">E45</font></i>: 
(  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">ax</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SA</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ax</font></span>)</span> &amp; <font color="Maroon">SA</font> <a href="mesfunc1.html#K4">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ax</font> holds <br/><font color="Maroon">SA</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">SA</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">ax</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="convfun1.html#D5">CONVFUN1:def 5</a></i>;<br/>
<a NAME="E10:19_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ax</font></span>)</span> &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">ax</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:19_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ax</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> )
 ;<br/>

<a NAME="E2:19_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">ax</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:19_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="prob_2.html#D3">PROB_2:def 3</a></i>;<br/>
<a NAME="E4:19_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E5:19_1_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="mesfunc3.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:19_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i><a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:19_1_1"/>
<font color="Maroon">ax</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:19_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ax</font></span>)</span> &amp; <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">ax</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:19_1_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ax</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 ;<br/>

<a NAME="E2:19_1_2"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K10">1.</a> 
 
<b>by </b><i>, <a class="ref" href="mesfunc3.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a>, <a class="ref" href="mesfunc1.html#D8">MESFUNC1:def 8</a></i>;<br/>
<b>then </b><font color="Maroon">ax</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span> <a href="extreal1.html#K2">*</a> <a href="mesfunc1.html#K10">1.</a> 
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="extreal2.html#T4">EXTREAL2:4</a></i>
;<br/>
<b>hence </b><a NAME="E4:19_1_2"/>
<font color="Maroon">ax</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="mesfunc3.html#T1" target="_self">Th1</a>, , </i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ax</font> implies ( ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> implies <font color="Maroon">SA</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> implies <font color="Maroon">SA</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> ) ) );<br/>
<a NAME="E12:19_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:19_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:19_1_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>]
 ;<br/>

<b>assume </b><a NAME="E2:19_1_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ax</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_3_1_1"/>
( <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> or <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_3_1_1_1"><b>case </b></a><a NAME="E1:19_1_3_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:19_1_3_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:19_1_3_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ax</font> )
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:19_1_3_1_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:19_1_3_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ax</font>
 <b>by </b><i>, <a class="txt" href="mesfunc3.html#E1:19"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:19_1_3_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ax</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:19_1_3_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ax</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><font color="Maroon">SA</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<a href="supinf_2.html#K1">0.</a>  <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">ax</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , , , <a class="txt" href="mesfunc3.html#E2:19"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>
<br/>.= 
<font color="Maroon">ax</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
<br/>.= 
 <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i>, , <a class="txt" href="mesfunc3.html#E3:19"><i><font color="Green">E37</font></i></a></i>
;<br/><b>hence </b><a NAME="E9:19_1_3_1_1_1"/>
( ( <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> implies <font color="Maroon">SA</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) &amp; ( <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> implies <font color="Maroon">SA</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_3_1_1_2"><b>case </b></a><a NAME="E1:19_1_3_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_3_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_3_1_1_2_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_3_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:19_1_3_1_1_2_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><a NAME="E2:19_1_3_1_1_2_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:19_1_3_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ax</font>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><font color="Maroon">SA</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">SA</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">ax</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">ax</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , , <a class="txt" href="mesfunc3.html#E1:19"><i><font color="Green">E35</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_3_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:19_1_3_1_1_2_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><a NAME="E2:19_1_3_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:19_1_3_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ax</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:19_1_3_1_1_2_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:19_1_3_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ax</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>thus </b><font color="Maroon">SA</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">SA</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">ax</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="mesfunc3.html#E1:19"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K2">+</a> <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i>, , , , <a class="txt" href="mesfunc3.html#E2:19"><i><font color="Green">E36</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:19_1_3_1_1_2"/>
( ( <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> implies <font color="Maroon">SA</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) &amp; ( <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> implies <font color="Maroon">SA</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:19_1_3"/>
( ( <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> implies <font color="Maroon">SA</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) &amp; ( <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> implies <font color="Maroon">SA</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> ) )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E14:19_1"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<a NAME="E15:19_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">ax</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E16:19_1"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">ax</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:19"/>
(  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">ax</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">a</font> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">ax</font> holds <br/><font color="Maroon">ax</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> ) &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">ax</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div>
