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

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">r</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:4"/><i><font color="Green" title="E4">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green" title="E5">A2</font></i>: 
for <font color="Olive" title="b1">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>   st 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">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="c2">r</font> holds <br/><font color="Maroon" title="c2">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_1"/>
<font color="Maroon" title="c2">r</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><a NAME="E2:4_1"/><b>then </b>
<font color="Maroon" title="c2">r</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</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">L</font>
 ;<br/><b>hence </b><a NAME="E3:4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c4">r1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:4"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>and </b><br/><a NAME="E6:4"/><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">r1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T111" title="FINSEQ_3:th.111">FINSEQ_3:111</a></i>;<br/>
<a NAME="E7:4"/><i><font color="Green" title="E8">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E8:4"/><b>then </b><i><font color="Green" title="E9">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">r1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="uproots.html#E6:4"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_2"/>
<font color="Maroon" title="c4">r1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><a NAME="E2:4_2"/><b>then </b>
<font color="Maroon" title="c4">r1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</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">L</font>
 ;<br/><a NAME="E3:4_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></i>;<br/><b>hence </b><a NAME="E4:4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="uproots.html#E8:4"><i><font color="Green" title="E9">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c5">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c6">r2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E11:4"/><i><font color="Green" title="E10">A7</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>and </b><br/><a NAME="E12:4"/><i><font color="Green" title="E11">A8</font></i>: 
<font color="Maroon" title="c4">r1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">b</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">r2</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T111" title="FINSEQ_3:th.111">FINSEQ_3:111</a></i>;<br/>
<a NAME="E13:4"/><i><font color="Green" title="E12">A9</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">b</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<div><i><font color="Green" title="E13">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">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="E1:4_3"/><i><font color="Green" title="E14">A11</font></i>: 
<font color="Maroon" title="c7">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="c6">r2</font>
 ;<br/><a NAME="E2:4_3"/><i><font color="Green" title="E15">A12</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">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">r1</font>
 <b>by </b><i><a class="txt" href="uproots.html#E12:4"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="uproots.html#E13:4"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="uproots.html#E1:4_3"><i><font color="Green" title="E14">A11</font></i></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a></i>;<br/><a NAME="E3:4_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:4_3"><i><font color="Green" title="E14">A11</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:4_3"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:4_3"/><b>then </b><i><font color="Green" title="E16">A13</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E6:4_3"/><i><font color="Green" title="E17">A14</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">i</font></span>)</span> <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="c2">r</font>
 <b>by </b><i><a class="txt" href="uproots.html#E6:4"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="uproots.html#E7:4"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="uproots.html#E2:4_3"><i><font color="Green" title="E15">A12</font></i></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a></i>;<br/><b>thus </b><font color="Maroon" title="c6">r2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> = 
<font color="Maroon" title="c4">r1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">i</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E12:4"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="uproots.html#E13:4"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="uproots.html#E1:4_3"><i><font color="Green" title="E14">A11</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E6:4"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="uproots.html#E7:4"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="uproots.html#E2:4_3"><i><font color="Green" title="E15">A12</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="uproots.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="uproots.html#E5:4_3"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="uproots.html#E6:4_3"><i><font color="Green" title="E17">A14</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E15:4"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r</font>
 
<b>by </b><i><a class="txt" href="uproots.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:4"/><b>then </b><i><font color="Green" title="E14">A15</font></i>: 
1 <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="c2">r</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E17:4"/><i><font color="Green" title="E15">A16</font></i>: 
2 <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="c2">r</font>
 
<b>by </b><i><a class="txt" href="uproots.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<i><font color="Green" title="E16">A17</font></i>: <font color="Maroon" title="c2">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<font color="Maroon" title="c4">r1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="uproots.html#E6:4"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="uproots.html#E7:4"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="finseq_1.html#T37" title="FINSEQ_1:th.37">FINSEQ_1:37</a></i>
<br/>.= 
<font color="Maroon" title="c4">r1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1

;<br/>
<b>thus </b> <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c2">r</font> = 
<font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c4">r1</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E6:4"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="fvsum_1.html#T89" title="FVSUM_1:th.89">FVSUM_1:89</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c6">r2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E12:4"><i><font color="Green" title="E11">A8</font></i></a>, <a class="ref" href="fvsum_1.html#T89" title="FVSUM_1:th.89">FVSUM_1:89</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E14:4"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="polynom3.html#T1" title="POLYNOM3:th.1">POLYNOM3:1</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="txt" href="uproots.html#E5:4"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="uproots.html#E16:4"><i><font color="Green" title="E14">A15</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E11:4"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="uproots.html#E17:4"><i><font color="Green" title="E15">A16</font></i></a>, <a class="txt" href="uproots.html#E18:4"><i><font color="Green" title="E16">A17</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/>


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