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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="vectsp_1.html#V5">left-distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">z0</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">z1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>consider </b><font color="Maroon">F</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="E2:53"/><i><font color="Green">E33</font></i>: 
 <a href="polynom4.html#K2">eval</a> <span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<font color="Maroon">z1</font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E3:53"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<font color="Maroon">z1</font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span>
 <b>and </b><br/><a NAME="E4:53"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<font color="Maroon">z1</font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom4.html#D2">POLYNOM4:def 2</a></i>;<br/>
<a NAME="E5:53"/><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#R1">&lt;=</a> 2
 
<b>by </b><i><a class="ref" href="polynom5.html#T3" target="_self">Th3</a>, <a class="ref" href="polynom5.html#T2" target="_self">Th2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i>, <a class="ref" href="cqc_the1.html#T3">CQC_THE1:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>suppose </b></a><a NAME="E1:53_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:53_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<font color="Maroon">z1</font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="polynom5.html#T3" target="_self">Th3</a>, <a class="ref" href="polynom4.html#T8">POLYNOM4:8</a></i>;<br/><b>hence </b> <a href="polynom4.html#K2">eval</a> <span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<font color="Maroon">z1</font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span>,<font color="Maroon">x</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="polynom4.html#T20">POLYNOM4:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon">z0</font>
<b>by </b><i>, <a class="txt" href="polynom5.html#E3:53"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<font color="Maroon">z0</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
<br/>.= 
<font color="Maroon">z0</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
<br/>.= 
<font color="Maroon">z0</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon">z0</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="txt" href="polynom5.html#E3:53"><i><font color="Green">E34</font></i></a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2"><b>suppose </b></a><a NAME="E1:53_1_2"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:53_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</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">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/><a NAME="E3:53_1_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<font color="Maroon">z1</font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<font color="Maroon">z1</font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<font color="Maroon">z1</font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,0</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<font color="Maroon">z0</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,0</span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E3:53"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<font color="Maroon">z0</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a></i>
<br/>.= 
<font color="Maroon">z0</font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/><a NAME="E5:53_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">z0</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>hence </b> <a href="polynom4.html#K2">eval</a> <span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<font color="Maroon">z1</font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span>,<font color="Maroon">x</font> = 
<font color="Maroon">z0</font>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
<br/>.= 
<font color="Maroon">z0</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
<br/>.= 
<font color="Maroon">z0</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
<br/>.= 
<font color="Maroon">z0</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<font color="Maroon">z1</font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T3" target="_self">Th3</a>, , <a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></i>
<br/>.= 
<font color="Maroon">z0</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E3:53"><i><font color="Green">E34</font></i></a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_3"><b>suppose </b></a><a NAME="E1:53_1_3"/><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:53_1_3"/><b>then </b><i><font color="Green">E47</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E48</font></i>: <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<font color="Maroon">z1</font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<font color="Maroon">z1</font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<font color="Maroon">z1</font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,0</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<font color="Maroon">z0</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,0</span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E3:53"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<font color="Maroon">z0</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a></i>
<br/>.= 
<font color="Maroon">z0</font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/><a NAME="E4:53_1_3"/><i><font color="Green">E49</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 2 = 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<font color="Maroon">z1</font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">2 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default">2 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,1</span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E2:53_1_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="polynom5.html#E3:53"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="group_1.html#T98">GROUP_1:98</a></i>
;<br/><a NAME="E6:53_1_3"/><b>then </b>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">z0</font>,<span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>by </b><i>, <a class="txt" href="polynom5.html#E1:53_1_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E7:53_1_3"/>
 <a href="polynom4.html#K2">eval</a> <span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">z0</font>,<font color="Maroon">z1</font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z0</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T62">RLVECT_1:62</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
