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

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:43_1_2"/><i><font color="Green">E36</font></i>: 
 ex <font color="Olive">y1</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">y1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">y1</font> &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 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">y1</font> holds <br/><font color="Olive">y1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="polynom2.html#K4" target="_self">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:43_1_2"/><i><font color="Green">E37</font></i>: 
 ex <font color="Olive">y2</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">y2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">y2</font> &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 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">y2</font> holds <br/><font color="Olive">y2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="polynom2.html#K4" target="_self">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span> ) )
 ;<br/>

<b>consider </b><font color="Maroon">y1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:43_1_2"/><i><font color="Green">E39</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">y1</font> &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 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y1</font> holds <br/><font color="Maroon">y1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="polynom2.html#K4" target="_self">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:43_1_2"/><i><font color="Green">E42</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">y2</font> &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 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y2</font> holds <br/><font color="Maroon">y2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="polynom2.html#K4" target="_self">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span> ) )
 <b>by </b><i/>;<br/>
<a NAME="E7:43_1_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y1</font> holds <br/><font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:43_1_2_1"/><i><font color="Green">E45</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y1</font> )
 ;<br/>

<a NAME="E2:43_1_2_1"/><b>then </b>
( <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">y1</font></span>)</span> &amp; <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">y2</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:43_1_2_1"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">y1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="polynom2.html#K4" target="_self">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">y2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">y2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:43_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div>
