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

<b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<a NAME="E2:18"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<b>set </b><font color="Maroon">B</font> =  <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span>;<br/>
<a NAME="E3:18"/><i><font color="Green">E42</font></i>: 
 <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T30">MATRIX_2:30</a></i>;<br/>
<a NAME="E4:18"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="matrix_2.html#D17">MATRIX_2:def 17</a></i>;<br/>
<a NAME="E5:18"/>
 <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T23">MATRIX_2:23</a></i>;<br/>
<a NAME="E6:18"/><b>then </b>
 <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span>
 
;<br/>
<a NAME="E7:18"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="matrix_2.html#D13">MATRIX_2:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p0</font> =  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> ;<br/>
<a NAME="E9:18"/>
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> is <a href="matrix_2.html#V3">even</a>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T29">MATRIX_2:29</a></i>;<br/>
<a NAME="E10:18"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">p0</font> is <a href="matrix_2.html#V3">even</a>
 
;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="matrix_3.html#K11">Path_product</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">F</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font>;<br/>
<b>set </b><font color="Maroon">Y</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>;<br/>
<b>set </b><font color="Maroon">X</font> =  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="matrix_7.html#K2" target="_self">IFIN</a> <span class="p1">(<span class="default"><a href="newton.html#K4">idseq</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>;<br/>
<a NAME="E11:18"/><i><font color="Green">E46</font></i>: 
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>  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_1"/>
(  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> or  not  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1"><b>case </b></a><a NAME="E1:18_1_1_1_1"/>
 <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_1_1"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:18_1_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_2"><b>case </b></a><a NAME="E1:18_1_1_1_2"/>
not  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_1_2"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:18_1_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:18"/>
 ex <font color="Olive">f2</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> st <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>  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span> holds <br/><font color="Olive">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 
<b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>();<br/></i>
<b>then consider </b><font color="Maroon">G0</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E14:18"/><i><font color="Green">E47</font></i>: 
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>  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span> holds <br/><font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> =  <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="newton.html#K4">idseq</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>;<br/>
<a NAME="E15:18"/><i><font color="Green">E52</font></i>: 
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>  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 
;<br/>
<a NAME="E16:18"/>
 ex <font color="Olive">f2</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> st <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>  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font> holds <br/><font color="Olive">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>)
 
<b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>();<br/></i>
<b>then consider </b><font color="Maroon">f2</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E18:18"/><i><font color="Green">E53</font></i>: 
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>  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>)
 ;<br/>
<a NAME="E19:18"/><i><font color="Green">E54</font></i>: 
(  <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 
<b>by </b><i>, <a class="ref" href="matrix_2.html#D17">MATRIX_2:def 17</a></i>;<br/>
<a NAME="E20:18"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E21:18"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E22:18"/>
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>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:18_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:18_2"/>
for <font color="Olive">p</font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font> holds  <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Olive">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>;<br/>

<a NAME="E1:18_2_1"/><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>;<br/>
<a NAME="E2:18_2_1"/>
1 <a href="finsop_1.html#K1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T73">FINSEQ_2:73</a></i>;<br/>
<a NAME="E3:18_2_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>;<br/>
<a NAME="E4:18_2_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:18_2_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:18_2_1_1"/><i><font color="Green">E61</font></i>: 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 
<b>by </b><i><a class="ref" href="fvsum_1.html#T11">FVSUM_1:11</a></i>;<br/>
<a NAME="E3:18_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T74">FINSEQ_2:74</a></i>;<br/>
<b>then </b>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span></span>)</span> = 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i><a class="ref" href="finsop_1.html#T5">FINSOP_1:5</a>, <a class="txt" href="matrix_7.html#E1"><i><font color="Green">Lemma34</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E5:18_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:18_2_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E6:18_2_1"/>
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E7:18_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
;<br/>
<a NAME="E8:18_2_1"/><b>then </b><i><font color="Green">E63</font></i>: 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_2_1"/>
( <font color="Maroon">p</font> is <a href="matrix_2.html#V3">even</a> or  not <font color="Maroon">p</font> is <a href="matrix_2.html#V3">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_1"><b>case </b></a><a NAME="E1:18_2_1_2_1_1"/>
<font color="Maroon">p</font> is <a href="matrix_2.html#V3">even</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_2_1_1"/>
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="matrix_2.html#D16">MATRIX_2:def 16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2_1_2"><b>case </b></a><a NAME="E1:18_2_1_2_1_2"/>
not <font color="Maroon">p</font> is <a href="matrix_2.html#V3">even</a>
 ;<br/><div class="add"><b>then </b> <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>,<font color="Maroon">p</font> = 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_2.html#D16">MATRIX_2:def 16</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T25">RLVECT_1:25</a></i>
;<br/><b>hence </b><a NAME="E3:18_2_1_2_1_2"/>
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E10:18_2_1"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_3_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> or <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_1"><b>case </b></a><a NAME="E1:18_2_1_3_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><i><font color="Green">E66</font></i>: <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">p</font>)
<b>by </b><i/>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
<b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p0</font> as    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/><a NAME="E4:18_2_1_3_1_1"/>
 <a href="matrix_2.html#K11">len</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="matrix_2.html#T20">MATRIX_2:20</a></i>;<br/><a NAME="E5:18_2_1_3_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="matrix_2.html#D16">MATRIX_2:def 16</a></i>;<br/><a NAME="E6:18_2_1_3_1_1"/>
for <font color="Olive">i</font>, <font color="Olive">j</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#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span></span>)</span> &amp; <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_1_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>let </b><font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:18_2_1_3_1_1_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span></span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 ;<br/>

<a NAME="E2:18_2_1_3_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E3:18_2_1_3_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E4:18_2_1_3_1_1_2"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a>, <a class="txt" href="matrix_7.html#E2:18"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E5:18_2_1_3_1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E1:18"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E6:18_2_1_3_1_1_2"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a>, </i>;<br/>
<a NAME="E7:18_2_1_3_1_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E2:18"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:18_2_1_3_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E1:18"><i><font color="Green">E39</font></i></a>, <a class="txt" href="matrix_7.html#E2:18"><i><font color="Green">E40</font></i></a>, </i>;<br/>
<a NAME="E9:18_2_1_3_1_1_2"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">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="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E4:18"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>;<br/>
<a NAME="E10:18_2_1_3_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E11:18_2_1_3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D12">MATRIX_1:def 12</a>, <a class="txt" href="matrix_7.html#E3:18"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:18_2_1_3_1_1"/>
<font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="matrix_7.html#T1" target="_self">Th1</a>, <a class="txt" href="matrix_7.html#E1"><i><font color="Green">Lemma34</font></i></a>, , <a class="ref" href="matrix_3.html#D7">MATRIX_3:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2"><b>case </b></a><a NAME="E1:18_2_1_3_1_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:18_2_1_3_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">len</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/><a NAME="E3:18_2_1_3_1_2"/><b>then </b><i><font color="Green">E100</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="matrix_2.html#T20">MATRIX_2:20</a></i>;<br/><a NAME="E4:18_2_1_3_1_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="newton.html#K4">idseq</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T54">FINSEQ_2:54</a></i>;<br/><b>then consider </b><font color="Maroon">i0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:18_2_1_3_1_2"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">i0</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="newton.html#K4">idseq</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i0</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><a NAME="E7:18_2_1_3_1_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">i0</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="matrix_7.html#T1" target="_self">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon">i0</font> = <font color="Maroon">i0</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="matrix_7.html#E1:18"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E9:18_2_1_3_1_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font>
 <b>by </b><i>, <a class="ref" href="matrix_7.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_2.html#T56">FINSEQ_2:56</a></i>;<br/><a NAME="E10:18_2_1_3_1_2"/>
<font color="Maroon">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/><a NAME="E11:18_2_1_3_1_2"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a>, <a class="txt" href="matrix_7.html#E1:18"><i><font color="Green">E39</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">j0</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i0</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><i><font color="Green">E105</font></i>: <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">p</font>)
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><a NAME="E14:18_2_1_3_1_2"/><i><font color="Green">E106</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</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#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> ) )
 <b>by </b><i><a class="ref" href="matrix_3.html#D7">MATRIX_3:def 7</a></i>;<br/><a NAME="E15:18_2_1_3_1_2"/><b>then </b>
<font color="Maroon">i0</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:18"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E16:18_2_1_3_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i0</font>,<font color="Maroon">j0</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E4:18"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E17:18_2_1_3_1_2"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a>, </i>;<br/><a NAME="E18:18_2_1_3_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i0</font>,<font color="Maroon">j0</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:18"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E19:18_2_1_3_1_2"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E7:18"><i><font color="Green">E44</font></i></a>, <a class="ref" href="matrix_1.html#D12">MATRIX_1:def 12</a>, <a class="txt" href="matrix_7.html#E2:18"><i><font color="Green">E40</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> implies  ex <font color="Olive">p3</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">p3</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 &amp; ( for <font color="Olive">n3</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">n3</font> &amp; <font color="Olive">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> holds <br/><font color="Olive">p3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n3</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) ) );<br/><a NAME="E20:18_2_1_3_1_2"/>
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> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="matrix_7.html#E4:18"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E21:18_2_1_3_1_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E22:18_2_1_3_1_2"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E23:18_2_1_3_1_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">d</font> = <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> <b>by </b><i><a class="txt" href="matrix_7.html#E11:18"><i><font color="Green">E46</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">q3</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="binarith.html#K11">*&gt;</a></span> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font> ;<br/><a NAME="E26:18_2_1_3_1_2"/><i><font color="Green">E110</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E27:18_2_1_3_1_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">q3</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E28:18_2_1_3_1_2"/>
for <font color="Olive">n3</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">n3</font> &amp; <font color="Olive">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">q3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">q3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n3</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n3</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:18_2_1_3_1_2_2"/><i><font color="Green">E112</font></i>: 
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n3</font> &amp; <font color="Maroon">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 ;<br/>

<a NAME="E2:18_2_1_3_1_2_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n3</font>
 
;<br/>
<b>hence </b><a NAME="E3:18_2_1_3_1_2_2"/>
<font color="Maroon">q3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">q3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n3</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E29:18_2_1_3_1_2"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 <b>by </b><i><a class="txt" href="matrix_7.html#E14:18"><i><font color="Green">E47</font></i></a>, <a class="ref" href="matrix_7.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E30:18_2_1_3_1_2"/><i><font color="Green">E117</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:18_2_1_3_1_2_3"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_3_1_2_3_1_1"/>
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_3_1_1_1"><b>case </b></a><a NAME="E1:18_2_1_3_1_2_3_1_1_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p3</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E3:18_2_1_3_1_2_3_1_1_1"/><i><font color="Green">E153</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 &amp; ( for <font color="Olive">n3</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">n3</font> &amp; <font color="Olive">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n3</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) );<br/><a NAME="E4:18_2_1_3_1_2_3_1_1_1"/><i><font color="Green">E155</font></i>: 
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>  <a href="numbers.html#K5">NAT</a>  holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_1_2_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:18_2_1_3_1_2_3_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_1_2_3_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_3_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_3_1_2_3_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_3_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:18_2_1_3_1_2_3_1_1_1_1_1_1_1"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">nx</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:18_2_1_3_1_2_3_1_1_1_1_1_1_1"/>
<font color="Maroon">nx</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p3</font>
 <b>by </b><i>, <a class="txt" href="matrix_7.html#E18:18"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:18_2_1_3_1_2_3_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p3</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:18_2_1_3_1_2_3_1_1_1_1_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p3</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> <b>by </b><i><a class="txt" href="matrix_7.html#E21:18"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E7:18_2_1_3_1_2_3_1_1_1_1_1_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> implies <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> ) &amp; ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> implies <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:18_2_1_3_1_2_3_1_1_1_1_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_3_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:18_2_1_3_1_2_3_1_1_1_1_1_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_3_1_2_3_1_1_1_1_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18_2_1_3_1_2_3_1_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 ;<br/>


</div><b>end;</b></div><a NAME="E5:18_2_1_3_1_2_3_1_1_1"/>
 ex <font color="Olive">f6</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> st <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>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Olive">f6</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="matrix_7.html#E19:18"><i><font color="Green">E54</font></i></a>);<br/></i><b>then consider </b><font color="Maroon">f6</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E7:18_2_1_3_1_2_3_1_1_1"/><i><font color="Green">E160</font></i>: 
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>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f6</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 ;<br/><a NAME="E8:18_2_1_3_1_2_3_1_1_1"/><i><font color="Green">E161</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E9:18_2_1_3_1_2_3_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="matrix_7.html#E15:18"><i><font color="Green">E52</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:18_2_1_3_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E4:18"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E11:18_2_1_3_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E12:18_2_1_3_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E13:18_2_1_3_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E162</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E21:18"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E14:18_2_1_3_1_2_3_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f6</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:18_2_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">e</font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f6</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>reconsider </b><font color="Maroon">q3</font> = <font color="Maroon">p3</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="binarith.html#K11">*&gt;</a></span> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font> ;<br/><i><font color="Green">E163</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q3</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p3</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="matrix_7.html#E18:18"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/><a NAME="E18:18_2_1_3_1_2_3_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E19:18_2_1_3_1_2_3_1_1_1"/><b>then </b>
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">p3</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E18:18"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E20:18_2_1_3_1_2_3_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p3</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E21:18_2_1_3_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E164</font></i>: 
<font color="Maroon">q3</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, <a class="txt" href="matrix_7.html#E18:18"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E22:18_2_1_3_1_2_3_1_1_1"/>
for <font color="Olive">n3</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">n3</font> &amp; <font color="Olive">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">q3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">q3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n3</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_1_2_3_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n3</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:18_2_1_3_1_2_3_1_1_1_3"/><i><font color="Green">E165</font></i>: 
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n3</font> &amp; <font color="Maroon">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_1_2_3_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_3_1_1_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="E1:18_2_1_3_1_2_3_1_1_1_3_1_1"/>
( <font color="Maroon">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">n3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_3_1_1_1_3_1_1_1"><b>case </b></a><a NAME="E1:18_2_1_3_1_2_3_1_1_1_3_1_1_1"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:18_2_1_3_1_2_3_1_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n3</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E18:18"><i><font color="Green">E53</font></i></a>, </i>;<br/><a NAME="E3:18_2_1_3_1_2_3_1_1_1_3_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:18_2_1_3_1_2_3_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">n3</font> <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">p3</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E18:18"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E5:18_2_1_3_1_2_3_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p3</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:18_2_1_3_1_2_3_1_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E168</font></i>: 
<font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E7:18_2_1_3_1_2_3_1_1_1_3_1_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n3</font>
 <b>by </b><i/>;<br/><a NAME="E8:18_2_1_3_1_2_3_1_1_1_3_1_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n3</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:18_2_1_3_1_2_3_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">n3</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">p3</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="matrix_7.html#E18:18"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E10:18_2_1_3_1_2_3_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">n3</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p3</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E11:18_2_1_3_1_2_3_1_1_1_3_1_1_1"/>
<font color="Maroon">q3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">q3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n3</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E8:18_2_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="matrix_7.html#E5:18_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_3_1_1_1_3_1_1_2"><b>case </b></a><a NAME="E1:18_2_1_3_1_2_3_1_1_1_3_1_1_2"/>
<font color="Maroon">n3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:18_2_1_3_1_2_3_1_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E169</font></i>: 
<font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:18_2_1_3_1_2_3_1_1_1_3_1_1_2"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:18_2_1_3_1_2_3_1_1_1_3_1_1_2"/>
<font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:18_2_1_3_1_2_3_1_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="matrix_7.html#E5:18_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:18_2_1_3_1_2_3_1_1_1_3_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E7:18_2_1_3_1_2_3_1_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E172</font></i>: 
<font color="Maroon">k</font> <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"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E8:18_2_1_3_1_2_3_1_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E173</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p3</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E18:18"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:18_2_1_3_1_2_3_1_1_1_3_1_1_2"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">f6</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="matrix_7.html#E10:18_2_1"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E10:18_2_1_3_1_2_3_1_1_1_3_1_1_2"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">q3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><font color="Maroon">q3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a>, , <a class="txt" href="matrix_7.html#E18:18"><i><font color="Green">E53</font></i></a>, <a class="txt" href="matrix_7.html#E3:18_2_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="matrix_7.html#E5:18_2_1"><i><font color="Green">E62</font></i></a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="txt" href="matrix_7.html#E8:18_2_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>
;<br/><b>hence </b><a NAME="E12:18_2_1_3_1_2_3_1_1_1_3_1_1_2"/>
<font color="Maroon">q3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">q3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n3</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="txt" href="matrix_7.html#E8:18_2_1"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18_2_1_3_1_2_3_1_1_1_3"/>
<font color="Maroon">q3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">q3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n3</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E23:18_2_1_3_1_2_3_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="matrix_7.html#E3:18_2_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="matrix_7.html#E4:18_2_1"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_3_1_1_2"><b>case </b></a><a NAME="E1:18_2_1_3_1_2_3_1_1_2"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_3_1_2_3_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18_2_1_3_1_2_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div><a NAME="E31:18_2_1_3_1_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i><a NAME="E32:18_2_1_3_1_2"/><b>then </b><i><font color="Green">E176</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1]
 ;<br/><a NAME="E33:18_2_1_3_1_2"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E34:18_2_1_3_1_2"/><b>then </b><i><font color="Green">E178</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E35:18_2_1_3_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E36:18_2_1_3_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>then consider </b><font color="Maroon">p3</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E38:18_2_1_3_1_2"/><i><font color="Green">E179</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 &amp; ( for <font color="Olive">n3</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">n3</font> &amp; <font color="Olive">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n3</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i>, <a class="txt" href="matrix_7.html#E15:18"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E39:18_2_1_3_1_2"/><i><font color="Green">E180</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E18:18"><i><font color="Green">E53</font></i></a>, <a class="txt" href="matrix_7.html#E19:18"><i><font color="Green">E54</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">A</font> =  <a href="numbers.html#K5">NAT</a>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) );<br/><a NAME="E41:18_2_1_3_1_2"/><i><font color="Green">E182</font></i>: 
for <font color="Olive">x3</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>  ex <font color="Olive">y3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x3</font>,<font color="Olive">y3</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x3</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_1_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_3_1_2_4_1_1"/>
( <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> or  not <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_4_1_1_1"><b>case </b></a><a NAME="E1:18_2_1_3_1_2_4_1_1_1"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">x3</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">x3</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:18_2_1_3_1_2_4_1_1_1"/><i><font color="Green">E186</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p3</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E4:18_2_1_3_1_2_4_1_1_1"/>
<font color="Maroon">x3</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p3</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="txt" href="matrix_7.html#E1:18_2_1"><i><font color="Green">E57</font></i></a>, </i>;<br/><a NAME="E5:18_2_1_3_1_2_4_1_1_1"/><b>then </b>
<font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p3</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:18_2_1_3_1_2_4_1_1_1"/>
 ex <font color="Olive">y3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x3</font>,<font color="Olive">y3</font>]
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:18_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="matrix_7.html#E3:18_2_1"><i><font color="Green">E58</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_4_1_1_2"><b>case </b></a><a NAME="E1:18_2_1_3_1_2_4_1_1_2"/>
not <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_3_1_2_4_1_1_2"/>
 ex <font color="Olive">y3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x3</font>,<font color="Olive">y3</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1_3_1_2_4"/>
 ex <font color="Olive">y3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x3</font>,<font color="Olive">y3</font>]
 ;<br/>


</div><b>end;</b></div><a NAME="E42:18_2_1_3_1_2"/>
 ex <font color="Olive">f4</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> st <br/>for <font color="Olive">x2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x2</font>,<font color="Olive">f4</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x2</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="matrix_7.html#E21:18"><i><font color="Green">E56</font></i></a>);<br/></i><b>then consider </b><font color="Maroon">f4</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E44:18_2_1_3_1_2"/><i><font color="Green">E187</font></i>: 
for <font color="Olive">x4</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( ( <font color="Olive">x4</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> implies <font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x4</font> ) &amp; ( not <font color="Olive">x4</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> implies <font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x4</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 ;<br/><a NAME="E45:18_2_1_3_1_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E46:18_2_1_3_1_2"/><b>then </b><i><font color="Green">E189</font></i>: 
<font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:18_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="matrix_7.html#E19:18"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E47:18_2_1_3_1_2"/><i><font color="Green">E190</font></i>: 
for <font color="Olive">n3</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">n3</font> &amp; <font color="Olive">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> holds <br/><font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n3</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n3</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:18_2_1_3_1_2_5"/><i><font color="Green">E191</font></i>: 
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n3</font> &amp; <font color="Maroon">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:18_2_1_3_1_2_5"/><b>then </b>
<font color="Maroon">n3</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E3:18_2_1_3_1_2_5"/><b>then </b><i><font color="Green">E192</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n3</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:18_2_1_3_1_2_5"/><b>then </b><i><font color="Green">E193</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:18_2_1_3_1_2_5"/>
<font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:18_2_1_3_1_2_5"/><b>then </b>
<font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E4:18"><i><font color="Green">E43</font></i></a>, <a class="txt" href="matrix_7.html#E5:18_2_1"><i><font color="Green">E62</font></i></a></i>;<br/>
<a NAME="E7:18_2_1_3_1_2_5"/><b>then </b><i><font color="Green">E194</font></i>: 
<font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E1:18_2_1"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E8:18_2_1_3_1_2_5"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n3</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="matrix_7.html#E19:18"><i><font color="Green">E54</font></i></a>, <a class="txt" href="matrix_7.html#E4:18"><i><font color="Green">E43</font></i></a>, <a class="txt" href="matrix_7.html#E15:18"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E9:18_2_1_3_1_2_5"/>
<font color="Maroon">n3</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E4:18"><i><font color="Green">E43</font></i></a>, , </i>;<br/>
<b>hence </b><a NAME="E10:18_2_1_3_1_2_5"/>
<font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n3</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E8:18_2_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="matrix_7.html#E10:18_2_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="matrix_7.html#E1:18_2_1"><i><font color="Green">E57</font></i></a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>;<br/><a NAME="E48:18_2_1_3_1_2"/><i><font color="Green">E197</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i0</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:18"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E49:18_2_1_3_1_2"/><b>then </b><i><font color="Green">E198</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i0</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_1_2_6"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_3_1_2_6_1"/>
( <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 or <font color="Maroon">i0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_6_1_1"><b>case </b></a><a NAME="E1:18_2_1_3_1_2_6_1_1"/>
<font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:18_2_1_3_1_2_6_1_1"/><b>then </b>
<font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_3_1_2_6_1_1"/>
<font color="Maroon">S<sub>4</sub></font>[0]
 <b>by </b><i><a class="txt" href="matrix_7.html#E10:18"><i><font color="Green">E45</font></i></a>, <a class="txt" href="matrix_7.html#E3:18_2_1"><i><font color="Green">E58</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_6_1_2"><b>case </b></a><a NAME="E1:18_2_1_3_1_2_6_1_2"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">i0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:18_2_1_3_1_2_6_1_2"/><b>then </b>
<font color="Maroon">i0</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E3:18_2_1_3_1_2_6_1_2"/><b>then </b><i><font color="Green">E200</font></i>: 
<font color="Maroon">i0</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E4:18_2_1_3_1_2_6_1_2"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:18"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:18_2_1_3_1_2_6_1_2"/>
<font color="Maroon">i0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:18_2_1_3_1_2_6_1_2"/><b>then </b>
<font color="Maroon">i0</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:18_2_1_3_1_2_6_1_2"/><b>then </b><i><font color="Green">E202</font></i>: 
<font color="Maroon">i0</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E10:18_2_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="matrix_7.html#E4:18"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> ;<br/><a NAME="E9:18_2_1_3_1_2_6_1_2"/>
<font color="Maroon">i0</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a>, <a class="txt" href="matrix_7.html#E5:18_2_1"><i><font color="Green">E62</font></i></a></i>;<br/><a NAME="E10:18_2_1_3_1_2_6_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font>
 ;<br/><b>then </b><font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i0</font> = 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i0</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i0</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E8:18_2_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="matrix_7.html#E4:18_2_1"><i><font color="Green">E59</font></i></a>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E10:18"><i><font color="Green">E45</font></i></a>, <a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
;<br/><b>hence </b><a NAME="E12:18_2_1_3_1_2_6_1_2"/>
<font color="Maroon">S<sub>4</sub></font>[0]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E51:18_2_1_3_1_2"/><b>then </b><i><font color="Green">E203</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 ;<br/><a NAME="E52:18_2_1_3_1_2"/><i><font color="Green">E204</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_1_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:18_2_1_3_1_2_7"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:18_2_1_3_1_2_7"/><b>then </b><i><font color="Green">E205</font></i>: 
<font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 
;<br/>
<a NAME="E3:18_2_1_3_1_2_7"/><i><font color="Green">E206</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_1_2_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_3_1_2_7_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_7_1_1_1"><b>case </b></a><a NAME="E1:18_2_1_3_1_2_7_1_1_1"/><i><font color="Green">E207</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:18_2_1_3_1_2_7_1_1_1"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E3:18_2_1_3_1_2_7_1_1_1"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:18_2_1_3_1_2_7_1_1_1"/><i><font color="Green">E210</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:18_2_1_3_1_2_7_1_1_1"/><i><font color="Green">E211</font></i>: 
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> &amp; <font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="matrix_7.html#E18:18"><i><font color="Green">E53</font></i></a>, , , </i>;<br/><a NAME="E6:18_2_1_3_1_2_7_1_1_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:18_2_1_3_1_2_7_1_1_1"/><b>then </b>
<font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E8:18_2_1_3_1_2_7_1_1_1"/><b>then </b><i><font color="Green">E212</font></i>: 
<font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:18_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="matrix_7.html#E10:18_2_1"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E9:18_2_1_3_1_2_7_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E10:18_2_1_3_1_2_7_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <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> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="matrix_7.html#E4:18"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E11:18_2_1_3_1_2_7_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E12:18_2_1_3_1_2_7_1_1_1"/><b>then </b><i><font color="Green">E213</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E13:18_2_1_3_1_2_7_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> <b>by </b><i/>;<br/><a NAME="E15:18_2_1_3_1_2_7_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><b>then </b><font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E1:18_2_1"><i><font color="Green">E57</font></i></a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E19:18"><i><font color="Green">E54</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a>, </i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
;<br/><b>hence </b><a NAME="E17:18_2_1_3_1_2_7_1_1_1"/>
<font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_1_2_7_1_1_2"><b>case </b></a><a NAME="E1:18_2_1_3_1_2_7_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:18_2_1_3_1_2_7_1_1_2"/><b>then </b>
not <font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_3_1_2_7_1_1_2"/>
<font color="Maroon">f4</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:18_2_1"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:18_2_1_3_1_2_7"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div><a NAME="E53:18_2_1_3_1_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="matrix_7.html#E5:18_2_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="matrix_7.html#E8:18_2_1"><i><font color="Green">E63</font></i></a>);<br/></i><a NAME="E54:18_2_1_3_1_2"/><b>then </b><i><font color="Green">E214</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i0</font>]
 ;<br/><a NAME="E55:18_2_1_3_1_2"/>
<font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:18"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E56:18_2_1_3_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i0</font>
 <b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>hence </b><a NAME="E57:18_2_1_3_1_2"/>
<font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E3:18_2_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="matrix_7.html#E4:18_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="matrix_7.html#E10:18_2_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="matrix_7.html#E3:18"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="finsop_1.html#T3">FINSOP_1:3</a>, <a class="txt" href="matrix_7.html#E4:18"><i><font color="Green">E43</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E12:18_2_1"/>
<font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">p</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:18_2"/>
<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="matrix_3.html#D8">MATRIX_3:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:18"/><b>then </b><i><font color="Green">E215</font></i>: 
 <a href="matrix_3.html#K11">Path_product</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f2</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_3"/><i><font color="Green">E216</font></i>: 
 <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><b>thus </b><font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span></span>)</span> = 
 <a href="matrix_7.html#K2" target="_self">IFIN</a> <span class="p1">(<span class="default"><a href="newton.html#K4">idseq</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_2.html#K15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<a NAME="E25:18"/><b>then </b><i><font color="Green">E217</font></i>: 
<font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E26:18"/><i><font color="Green">E218</font></i>: 
for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>  st <font color="Olive">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> holds <br/><font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Olive">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>;<br/>

<b>assume </b><a NAME="E1:18_4"/>
<font color="Maroon">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font>
 ;<br/>

<a NAME="E2:18_4"/><b>then </b><i><font color="Green">E219</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>,<font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/>
<a NAME="E3:18_4"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="fvsum_1.html#T8">FVSUM_1:8</a></i>;<br/>
<a NAME="E4:18_4"/><b>then </b><i><font color="Green">E220</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>,<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/>
<a NAME="E5:18_4"/><i><font color="Green">E221</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finsub_1.html#T18">FINSUB_1:18</a></i>;<br/>
<a NAME="E6:18_4"/>
 <a href="matrix_7.html#K2" target="_self">IFIN</a> <span class="p1">(<span class="default"><a href="newton.html#K4">idseq</a> <font color="Maroon">n</font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:18_4"/>
<font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="txt" href="matrix_7.html#E1"><i><font color="Green">Lemma34</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:18"/><i><font color="Green">E222</font></i>: 
for <font color="Olive">x</font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font> holds  <font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_5_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_1_1_1"><b>case </b></a><a NAME="E1:18_5_1_1_1"/><i><font color="Green">E223</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><i><font color="Green">E224</font></i>: <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">x</font>)
<b>by </b><i/>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
<b>by </b><i><a class="txt" href="matrix_7.html#E1"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><a NAME="E3:18_5_1_1_1"/>
 <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E1"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:18_5_1_1_1"/><b>then </b>
 <a href="matrix_7.html#K2" target="_self">IFIN</a> <span class="p1">(<span class="default"><a href="newton.html#K4">idseq</a> <font color="Maroon">n</font></span>)</span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:18_5_1_1_1"/>
<font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_5_1_1_2"><b>case </b></a><a NAME="E1:18_5_1_1_2"/><i><font color="Green">E225</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><i><font color="Green">E226</font></i>: <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">x</font>)
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i><a class="txt" href="matrix_7.html#E1"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><a NAME="E3:18_5_1_1_2"/>
not  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E1"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:18_5_1_1_2"/><b>then </b>
 <a href="matrix_7.html#K2" target="_self">IFIN</a> <span class="p1">(<span class="default"><a href="newton.html#K4">idseq</a> <font color="Maroon">n</font></span>)</span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:18_5_1_1_2"/>
<font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_5"/>
<font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E28:18"/>
for <font color="Olive">B'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span>  st <font color="Olive">B'</font> <a href="tarski.html#R1">c=</a>  <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">x</font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="finsub_1.html#K2">\</a> <font color="Olive">B'</font> holds <br/><font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B'</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:18_6"/>
( <font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a>  <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>thus </b><a NAME="E2:18_6"/>
for <font color="Olive">x</font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="finsub_1.html#K2">\</a> <font color="Maroon">B'</font> holds <br/><font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>;<br/>

<b>assume </b><a NAME="E1:18_6_1"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="finsub_1.html#K2">\</a> <font color="Maroon">B'</font>
 ;<br/>

<div><i><font color="Green">E228</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_6_1_1"/><i><font color="Green">E229</font></i>: 
 <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><b>thus </b><font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
 <a href="matrix_7.html#K2" target="_self">IFIN</a> <span class="p1">(<span class="default"><a href="newton.html#K4">idseq</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="finsub_1.html#K1">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_6_1_2"/><i><font color="Green">E230</font></i>: 
not  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><b>thus </b><font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
 <a href="matrix_7.html#K2" target="_self">IFIN</a> <span class="p1">(<span class="default"><a href="newton.html#K4">idseq</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="finsub_1.html#K1">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<a NAME="E4:18_6_1"/><b>then </b><i><font color="Green">E231</font></i>: 
( (  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> implies <font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font> ) &amp; ( not  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> implies <font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E232</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_6_1_3"/><i><font color="Green">E233</font></i>: 
 <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font>
 ;<br/><b>thus </b><font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font> = 
 <a href="matrix_7.html#K2" target="_self">IFIN</a> <span class="p1">(<span class="default"><a href="newton.html#K4">idseq</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">B'</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_6_1_4"/><i><font color="Green">E234</font></i>: 
not  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font>
 ;<br/><b>thus </b><font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font> = 
 <a href="matrix_7.html#K2" target="_self">IFIN</a> <span class="p1">(<span class="default"><a href="newton.html#K4">idseq</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">B'</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<a NAME="E7:18_6_1"/><b>then </b><i><font color="Green">E235</font></i>: 
( (  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> implies <font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font> ) &amp; ( not  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> implies <font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 
<b>by </b><i><a class="ref" href="matrix_7.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E8:18_6_1"/><i><font color="Green">E236</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:18_6_1"/><i><font color="Green">E237</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="fvsum_1.html#T8">FVSUM_1:8</a></i>;<br/>
<div><i><font color="Green">E238</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_6_1_5"/>
 <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:18_6_1_5"/><b>then </b>
(  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> or  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:18_6_1_5"/><b>then </b><i><font color="Green">E239</font></i>: 
(  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> or  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_6_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_6_1_5_1_1"/>
(  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> or  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 </span><b>by </b><i><a class="txt" href="matrix_7.html#E3:18"><i><font color="Green">E42</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_6_1_5_1_1_1"><b>case </b></a><a NAME="E1:18_6_1_5_1_1_1"/><i><font color="Green">E240</font></i>: 
 <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font>
 ;<br/><div class="add"><a NAME="E2:18_6_1_5_1_1_1"/><b>then </b><i><font color="Green">E241</font></i>: 
<font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
 <b>by </b><i/>;<br/><a NAME="E3:18_6_1_5_1_1_1"/><i><font color="Green">E242</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E1"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:18_6_1_5_1_1_1"/><i><font color="Green">E243</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E4:18"><i><font color="Green">E43</font></i></a>, <a class="txt" href="matrix_7.html#E10:18"><i><font color="Green">E45</font></i></a></i>;<br/><span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">x</font>)
<b>by </b><i>, <a class="txt" href="matrix_7.html#E1:18"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i><a class="txt" href="matrix_7.html#E11:18"><i><font color="Green">E46</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><b>hence </b><a NAME="E6:18_6_1_5_1_1_1"/>
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E7:18"><i><font color="Green">E44</font></i></a>, <a class="txt" href="matrix_7.html#E2:18"><i><font color="Green">E40</font></i></a>, <a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_6_1_5_1_1_2"><b>case </b></a><a NAME="E1:18_6_1_5_1_1_2"/><i><font color="Green">E244</font></i>: 
 <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">x</font>)
<b>by </b><i>, <a class="txt" href="matrix_7.html#E1:18"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
<b>by </b><i><a class="txt" href="matrix_7.html#E4:18"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><b>hence </b><a NAME="E3:18_6_1_5_1_1_2"/>
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E1"><i><font color="Green">Lemma34</font></i></a>, , <a class="txt" href="matrix_7.html#E4:18"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, <a class="txt" href="matrix_7.html#E2:18"><i><font color="Green">E40</font></i></a>, <a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:18_6_1_5"/>
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_6_1_6"/><i><font color="Green">E245</font></i>: 
not  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:18_6_1_6"/><b>then </b><i><font color="Green">E246</font></i>: 
<font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:18_6_1_6"/>
not  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E3:18"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:18_6_1_6"/><b>then </b><i><font color="Green">E247</font></i>: 
not  <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">x</font>)
<b>by </b><i>, <a class="txt" href="matrix_7.html#E1:18"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i><a class="txt" href="matrix_7.html#E7:18"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><b>hence </b><a NAME="E6:18_6_1_6"/>
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E4:18"><i><font color="Green">E43</font></i></a>, <a class="txt" href="matrix_7.html#E2:18"><i><font color="Green">E40</font></i></a>, <a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:18_6_1"/>
<font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E29:18"/><b>then </b>
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="setwiseo.html#D3">SETWISEO:def 3</a>, , , , </i>;<br/>
<b>hence </b><a NAME="E30:18"/>
 <a href="matrix_3.html#K12">Det</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="matrix_3.html#D9">MATRIX_3:def 9</a></i>;<br/>


</div>
