<?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="vectsp_1.html#V7">distributive</a> <a href="vectsp_2.html#V2">domRing-like</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p</font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">q</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E3:34"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E4:34"/><i><font color="Green">E45</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="polynom4.html#T8">POLYNOM4:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">lp1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:34"/><i><font color="Green">E46</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lp1</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<a NAME="E7:34"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="polynom4.html#T8">POLYNOM4:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">lq1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:34"/><i><font color="Green">E47</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lq1</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">lp1</font> = <font color="Maroon">lp1</font>, <font color="Maroon">lq1</font> = <font color="Maroon">lq1</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/>
<a NAME="E11:34"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">lp1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> &amp; <font color="Maroon">q</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">lq1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> )
 
<b>by </b><i>, , <a class="ref" href="algseq_1.html#T25">ALGSEQ_1:25</a></i>;<br/>
<b>set </b><font color="Maroon">lpq</font> = <font color="Maroon">lp1</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">lq1</font>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E13:34"/><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">lp1</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">lq1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E14:34"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">q</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">lp1</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">lq1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E15:34"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">r</font> holds <br/><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">k</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"><font color="Maroon">q</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">lp1</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">lq1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Olive">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom3.html#D11">POLYNOM3:def 11</a></i>;<br/>
<a NAME="E16:34"/><i><font color="Green">E54</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">lp1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">lq1</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E17:34"/>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E18:34"/><b>then </b>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E19:34"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<i><font color="Green">E56</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">lp1</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">lq1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">lq1</font> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">lp1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>

<br/>.= 
<font color="Maroon">lq1</font>
<b>by </b><i>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:34_2"/><i><font color="Green">E57</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">r</font>
 <b>and </b><br/><a NAME="E2:34_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/><b>reconsider </b><font color="Maroon">k1</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="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E4:34_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k1</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"><font color="Maroon">q</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">lp1</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">lq1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#T1" target="_self">Th1</a>, </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_2_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1"><b>suppose </b></a><a NAME="E1:34_2_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:34_2_1_1"/><i><font color="Green">E60</font></i>: 
(  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">d</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="fsm_1.html#T1">FSM_1:1</a></i>;<br/><a NAME="E4:34_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">lp1</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">lq1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">lq1</font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i/>;<br/><b>then </b><i><font color="Green">E61</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">lp1</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">lq1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">lq1</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="uproots.html#E2:34"><i><font color="Green">E43</font></i></a></i>
<br/>.= 
<font color="Maroon">lq1</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">d</font>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/><a NAME="E6:34_2_1_1"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lq1</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="txt" href="uproots.html#E2:34"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</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"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:34"><i><font color="Green">E42</font></i></a>, <a class="txt" href="uproots.html#E3:34"><i><font color="Green">E44</font></i></a>, <a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_2"><b>suppose </b></a><a NAME="E1:34_2_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:34_2_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:34_2_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E4:34_2_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><b>hence </b><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<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> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">lp1</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">lq1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:34"><i><font color="Green">E42</font></i></a>, <a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then </b> <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">r</font> = 
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="uproots.html#T2" target="_self">Th2</a>, <a class="ref" href="matrix_3.html#T14">MATRIX_3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">lp1</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">lq1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="uproots.html#T1" target="_self">Th1</a>, <a class="ref" href="uproots.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">lp1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">lq1</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/>
<a NAME="E23:34"/><b>then </b>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E24:34"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="polynom3.html#T28">POLYNOM3:28</a></i>;<br/>


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