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

<b>let </b><font color="Maroon" title="c1">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c2">K</font>;<br/>





<b>assume </b><a NAME="E1:105"/><i><font color="Green" title="E92">A1</font></i>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">D</font> =  <a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">i</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1"/>
(  not <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span> or <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1"><b>suppose </b></a><a NAME="E1:105_1_1"/>
not <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:105_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T65" title="ZFMISC_1:th.65">ZFMISC_1:65</a></i>;<br/><b>then </b> <a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">i</font> = 
 <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix13.html#T51" target="_self" title="MATRIX13:th.51">TT51</a></i>
<br/>.= 
<font color="Maroon" title="c3">M</font>
<b>by </b><i><a class="ref" href="matrix13.html#T46" target="_self" title="MATRIX13:th.46">TT46</a></i>
;<br/><b>hence </b><a NAME="E4:105_1_1"/>
 <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c3">M</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2"><b>suppose </b></a><a NAME="E1:105_1_2"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c5">P</font>, <font color="Maroon" title="c6">Q</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:105_1_2"/><i><font color="Green" title="E93">A3</font></i>: 
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">Q</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">EqSegm</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def5</a></i>;<br/><a NAME="E4:105_1_2"/><i><font color="Green" title="E94">B0</font></i>: 
 <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">i</font>
 <b>by </b><i><a class="ref" href="matrix13.html#T51" target="_self" title="MATRIX13:th.51">TT51</a></i>;<br/><a NAME="E5:105_1_2"/>
( <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  iff <font color="Maroon" title="c6">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="matrix13.html#E3:105_1_2"><i><font color="Green" title="E93">A3</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">P2</font>, <font color="Maroon" title="c8">Q2</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:105_1_2"/><i><font color="Green" title="E95">A4</font></i>: 
( <font color="Maroon" title="c7">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c8">Q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font></span>)</span> )
 <b>and </b><br/><a NAME="E8:105_1_2"/>
( <font color="Maroon" title="c7">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c8">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c6">Q</font> )
 <b>and </b><br/><a NAME="E9:105_1_2"/>
(  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">P</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c8">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">Q</font> )
 <b>and </b><br/><a NAME="E10:105_1_2"/><i><font color="Green" title="E96">A5</font></i>: 
 <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">P2</font>,<font color="Maroon" title="c8">Q2</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:105_1_2"><i><font color="Green" title="E93">A3</font></i></a>, <a class="ref" href="matrix13.html#T57" target="_self" title="MATRIX13:th.57">TT57</a>, <a class="txt" href="matrix13.html#E4:105_1_2"><i><font color="Green" title="E94">B0</font></i></a></i>;<br/><a NAME="E11:105_1_2"/>
 <a href="matrix13.html#K5" title="MATRIX13:func.5">EqSegm</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:105_1_2"><i><font color="Green" title="E93">A3</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def4</a></i>;<br/><a NAME="E12:105_1_2"/><b>then </b><i><font color="Green" title="E97">A6</font></i>: 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:105_1_2"><i><font color="Green" title="E93">A3</font></i></a>, <a class="ref" href="matrix13.html#T83" target="_self" title="MATRIX13:th.83">TT83</a></i>;<br/><a NAME="E13:105_1_2"/><i><font color="Green" title="E98">B1</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><a NAME="E14:105_1_2"/><b>then </b><i><font color="Green" title="E99">B2</font></i>: 
( <font color="Maroon" title="c7">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E7:105_1_2"><i><font color="Green" title="E95">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E15:105_1_2"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">P2</font>,<font color="Maroon" title="c8">Q2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E7:105_1_2"><i><font color="Green" title="E95">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E16:105_1_2"/><b>then </b><i><font color="Green" title="E100">A7</font></i>: 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:105_1_2"><i><font color="Green" title="E93">A3</font></i></a>, <a class="txt" href="matrix13.html#E10:105_1_2"><i><font color="Green" title="E96">A5</font></i></a>, <a class="txt" href="matrix13.html#E12:105_1_2"><i><font color="Green" title="E97">A6</font></i></a>, <a class="ref" href="matrix13.html#T79" target="_self" title="MATRIX13:th.79">TT79</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">q</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E18:105_1_2"/><i><font color="Green" title="E101">A8</font></i>: 
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">q</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">EqSegm</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def5</a></i>;<br/><a NAME="E19:105_1_2"/>
not <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:105_1_2_1"/><i><font color="Green" title="E102">A9</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">p</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c11">i0</font> = <font color="Maroon" title="c1">i</font> as  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="pscomp_1.html#D1" title="PSCOMP_1:def.1">PSCOMP_1:def 1</a></i>;<br/>
<a NAME="E3:105_1_2_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">p</font>
 
<b>by </b><i><a class="txt" href="matrix13.html#E1:105_1_2_1"><i><font color="Green" title="E102">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">q1</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:105_1_2_1"/><i><font color="Green" title="E103">A10</font></i>: 
( <font color="Maroon" title="c12">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">q</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c12">q1</font> &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">EqSegm</a> <font color="Maroon" title="c3">M</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">i0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c12">q1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E18:105_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="ref" href="matrix13.html#T65" target="_self" title="MATRIX13:th.65">TT65</a></i>;<br/>
<a NAME="E6:105_1_2_1"/><i><font color="Green" title="E104">A11</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:105_1_2_1"/><i><font color="Green" title="E105">A12</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q1</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E5:105_1_2_1"><i><font color="Green" title="E103">A10</font></i></a>, <a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></i>;<br/>
<a NAME="E9:105_1_2_1"/>
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; not 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">q1</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="pscomp_1.html#D1" title="PSCOMP_1:def.1">PSCOMP_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">y</font> = <font color="Maroon" title="c13">y</font> as  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="matrix13.html#E8:105_1_2_1"><i><font color="Green" title="E105">A12</font></i></a></i>;<br/>
<a NAME="E11:105_1_2_1"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">q1</font>
 
<b>by </b><i><a class="txt" href="matrix13.html#E8:105_1_2_1"><i><font color="Green" title="E105">A12</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:105_1_2_1"/><b>then </b>
( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c10">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E5:105_1_2_1"><i><font color="Green" title="E103">A10</font></i></a>, <a class="txt" href="matrix13.html#E18:105_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="ref" href="matrix13.html#T67" target="_self" title="MATRIX13:th.67">TT67</a></i>;<br/>
<a NAME="E13:105_1_2_1"/><b>then </b><i><font color="Green" title="E106">A13</font></i>: 
( <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i0</font>,<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E1:105"><i><font color="Green" title="E92">A1</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="ref" href="finseq_2.html#T71" title="FINSEQ_2:th.71">FINSEQ_2:71</a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></i>;<br/>
 <a href="matrix13.html#K5" title="MATRIX13:func.5">EqSegm</a> <font color="Maroon" title="c3">M</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">i0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c12">q1</font> = 
 <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c3">M</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">i0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="txt" href="matrix13.html#E5:105_1_2_1"><i><font color="Green" title="E103">A10</font></i></a>, <a class="txt" href="matrix13.html#E8:105_1_2_1"><i><font color="Green" title="E105">A12</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def4</a></i>
<br/>.= 
<span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span>
<b>by </b><i><a class="txt" href="matrix13.html#E13:105_1_2_1"><i><font color="Green" title="E106">A13</font></i></a>, <a class="ref" href="matrix13.html#T44" target="_self" title="MATRIX13:th.44">TT44</a></i>
;<br/>
<b>hence </b><a NAME="E15:105_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="matrix13.html#E5:105_1_2_1"><i><font color="Green" title="E103">A10</font></i></a>, <a class="txt" href="matrix13.html#E6:105_1_2_1"><i><font color="Green" title="E104">A11</font></i></a>, <a class="ref" href="matrix_3.html#T36" title="MATRIX_3:th.36">MATRIX_3:36</a></i>;<br/>


</div><b>end;</b></div><a NAME="E20:105_1_2"/><b>then </b>
( <font color="Maroon" title="c9">p</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Maroon" title="c9">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E18:105_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="ref" href="matrix13.html#T67" target="_self" title="MATRIX13:th.67">TT67</a>, <a class="ref" href="zfmisc_1.html#T65" title="ZFMISC_1:th.65">ZFMISC_1:65</a></i>;<br/><a NAME="E21:105_1_2"/><b>then </b>
( <font color="Maroon" title="c9">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c10">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font></span>)</span> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E13:105_1_2"><i><font color="Green" title="E98">B1</font></i></a>, <a class="txt" href="matrix13.html#E14:105_1_2"><i><font color="Green" title="E99">B2</font></i></a>, <a class="txt" href="matrix13.html#E18:105_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">XBOOLE_1:33</a>, <a class="ref" href="matrix13.html#T67" target="_self" title="MATRIX13:th.67">TT67</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E22:105_1_2"/><b>then </b><i><font color="Green" title="E102">A9</font></i>: 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E4:105_1_2"><i><font color="Green" title="E94">B0</font></i></a>, <a class="ref" href="matrix13.html#T80" target="_self" title="MATRIX13:th.80">TT80</a></i>;<br/><a NAME="E23:105_1_2"/>
 <a href="matrix13.html#K5" title="MATRIX13:func.5">EqSegm</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E18:105_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def4</a></i>;<br/><a NAME="E24:105_1_2"/><b>then </b>
 <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">p</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E18:105_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="ref" href="matrix13.html#T83" target="_self" title="MATRIX13:th.83">TT83</a></i>;<br/><b>hence </b><a NAME="E25:105_1_2"/>
 <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E16:105_1_2"><i><font color="Green" title="E100">A7</font></i></a>, <a class="txt" href="matrix13.html#E18:105_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="matrix13.html#E22:105_1_2"><i><font color="Green" title="E102">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
