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<b>let </b><font color="Maroon" title="c1">n'</font>, <font color="Maroon" title="c2">m'</font>, <font color="Maroon" title="c3">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c4">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">K</font>;<br/><b>let </b><font color="Maroon" title="c6">M'</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n'</font>,<font color="Maroon" title="c2">m'</font>,<font color="Maroon" title="c4">K</font>;<br/>











<b>assume </b><a NAME="E1:105"/><i><font color="Green" title="E91">A1</font></i>: 
<font color="Maroon" title="c5">a</font> <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="c4">K</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c7">L</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>;<br/>
<b>set </b><font color="Maroon" title="c8">aL</font> = <font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">R</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c10">P</font>, <font color="Maroon" title="c11">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"/><i><font color="Green" title="E92">A2</font></i>: 
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">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="c6">M'</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">Q</font> )
 <b>and </b><br/><a NAME="E4:105"/><i><font color="Green" title="E93">A3</font></i>: 
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c6">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#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">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="c4">K</font> )
 <b>by </b><i><a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></i>;<br/>
<a NAME="E5:105"/><i><font color="Green" title="E94">A4</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">M'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></i>;<br/>
<div><i><font color="Green" title="E95">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_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_1"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> or  not <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_1"><b>suppose </b></a><a NAME="E1:105_1_1_1"/><i><font color="Green" title="E96">A6</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c12">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:105_1_1_1"/><i><font color="Green" title="E97">A7</font></i>: 
<font color="Maroon" title="c10">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> <font color="Maroon" title="c12">k</font>
 <b>by </b><i><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">Th43</a></i>;<br/><a NAME="E4:105_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c10">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:105_1_1_1"><i><font color="Green" title="E97">A7</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:105_1_1_1"/><i><font color="Green" title="E98">A8</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c10">P</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c10">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:105_1_1_1"><i><font color="Green" title="E96">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:105_1_1_1"/><i><font color="Green" title="E99">A9</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c10">P</font></span>)</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:105_1_1_1"><i><font color="Green" title="E97">A7</font></i></a>, <a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">x</font> = <font color="Maroon" title="c13">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="matrix13.html#E6:105_1_1_1"><i><font color="Green" title="E98">A8</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c15">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:105_1_1_1"/><i><font color="Green" title="E100">A10</font></i>: 
<font color="Maroon" title="c11">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> <font color="Maroon" title="c15">n</font>
 <b>by </b><i><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">Th43</a></i>;<br/><a NAME="E11:105_1_1_1"/><i><font color="Green" title="E101">A11</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c11">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Q</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E10:105_1_1_1"><i><font color="Green" title="E100">A10</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E12:105_1_1_1"/><i><font color="Green" title="E102">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M'</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><a NAME="E13:105_1_1_1"/><b>then </b><i><font color="Green" title="E103">A13</font></i>: 
( <font color="Maroon" title="c11">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="c6">M'</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M'</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E3:105"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="matrix13.html#T67" target="_self" title="MATRIX13:th.67">Th67</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:105_1_1_1"/><b>then </b>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span>,<font color="Maroon" title="c14">x</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="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c11">Q</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E6:105_1_1_1"><i><font color="Green" title="E98">A8</font></i></a>, <a class="txt" href="matrix13.html#E7:105_1_1_1"><i><font color="Green" title="E99">A9</font></i></a>, <a class="ref" href="matrix13.html#T47" target="_self" title="MATRIX13:th.47">Th47</a></i>;<br/><a NAME="E15:105_1_1_1"/><b>then </b><i><font color="Green" title="E104">A14</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span>,<font color="Maroon" title="c14">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c11">Q</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E11:105_1_1_1"><i><font color="Green" title="E101">A11</font></i></a>, <a class="txt" href="matrix13.html#E13:105_1_1_1"><i><font color="Green" title="E103">A13</font></i></a>, <a class="ref" href="matrix13.html#T87" target="_self" title="MATRIX13:th.87">Th87</a>, <a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>;<br/> <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span>,<font color="Maroon" title="c14">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span>,<font color="Maroon" title="c14">x</font></span>)</span></span>)</span> = 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span>,<font color="Maroon" title="c14">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span>,<font color="Maroon" title="c14">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E3:105"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></i>
<br/>.= 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span>,<font color="Maroon" title="c14">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span>,<font color="Maroon" title="c14">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E3:105"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></i>
<br/>.= 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font>
<b>by </b><i><a class="txt" href="matrix13.html#E3:105"><i><font color="Green" title="E92">A2</font></i></a>, <a class="txt" href="matrix13.html#E6:105_1_1_1"><i><font color="Green" title="E98">A8</font></i></a>, <a class="txt" href="matrix13.html#E12:105_1_1_1"><i><font color="Green" title="E102">A12</font></i></a>, <a class="txt" href="matrix13.html#E15:105_1_1_1"><i><font color="Green" title="E104">A14</font></i></a>, <a class="ref" href="matrix13.html#T59" target="_self" title="MATRIX13:th.59">Th59</a></i>
<br/>.= 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font>
<b>by </b><i><a class="txt" href="matrix13.html#E3:105"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></i>
;<br/><a NAME="E17:105_1_1_1"/><b>then </b>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E6:105_1_1_1"><i><font color="Green" title="E98">A8</font></i></a>, <a class="txt" href="matrix13.html#E7:105_1_1_1"><i><font color="Green" title="E99">A9</font></i></a>, <a class="ref" href="matrix11.html#T35" title="MATRIX11:th.35">MATRIX11:35</a></i>;<br/><a NAME="E18:105_1_1_1"/><b>then </b>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">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="c4">K</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E1:105"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="matrix13.html#E4:105"><i><font color="Green" title="E93">A3</font></i></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>;<br/><b>hence </b><a NAME="E19:105_1_1_1"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c6">M'</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:105"><i><font color="Green" title="E92">A2</font></i></a>, <a class="txt" href="matrix13.html#E4:105"><i><font color="Green" title="E93">A3</font></i></a>, <a class="txt" href="matrix13.html#E5:105"><i><font color="Green" title="E94">A4</font></i></a>, <a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_2"><b>suppose </b></a><a NAME="E1:105_1_1_2"/><i><font color="Green" title="E96">A15</font></i>: 
not <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font>
 ;<br/><div class="add"> <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font> = 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font>
<b>by </b><i><a class="txt" href="matrix13.html#E3:105"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></i>
<br/>.= 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font>
<b>by </b><i><a class="txt" href="matrix13.html#E3:105"><i><font color="Green" title="E92">A2</font></i></a>, <a class="txt" href="matrix13.html#E1:105_1_1_2"><i><font color="Green" title="E96">A15</font></i></a>, <a class="ref" href="matrix13.html#T60" target="_self" title="MATRIX13:th.60">Th60</a></i>
<br/>.= 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font>
<b>by </b><i><a class="txt" href="matrix13.html#E3:105"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></i>
;<br/><b>hence </b><a NAME="E3:105_1_1_2"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c6">M'</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:105"><i><font color="Green" title="E92">A2</font></i></a>, <a class="txt" href="matrix13.html#E4:105"><i><font color="Green" title="E93">A3</font></i></a>, <a class="txt" href="matrix13.html#E5:105"><i><font color="Green" title="E94">A4</font></i></a>, <a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c12">P1</font>, <font color="Maroon" title="c13">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="E8:105"/><i><font color="Green" title="E96">A16</font></i>: 
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</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="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">Q1</font> )
 <b>and </b><br/><a NAME="E9:105"/><i><font color="Green" title="E97">A17</font></i>: 
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></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#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">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="c4">K</font> )
 <b>by </b><i><a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_2_1"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">P1</font> or  not <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">P1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_1"><b>suppose </b></a><a NAME="E1:105_2_1_1"/><i><font color="Green" title="E98">A18</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">P1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c14">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:105_2_1_1"/><i><font color="Green" title="E99">A19</font></i>: 
<font color="Maroon" title="c12">P1</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> <font color="Maroon" title="c14">k</font>
 <b>by </b><i><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">Th43</a></i>;<br/><a NAME="E4:105_2_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c12">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">P1</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:105_2_1_1"><i><font color="Green" title="E99">A19</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:105_2_1_1"/><i><font color="Green" title="E100">A20</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c12">P1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c12">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:105_2_1_1"><i><font color="Green" title="E98">A18</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:105_2_1_1"/><i><font color="Green" title="E101">A21</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c12">P1</font></span>)</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">P1</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:105_2_1_1"><i><font color="Green" title="E99">A19</font></i></a>, <a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">x</font> = <font color="Maroon" title="c15">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="matrix13.html#E6:105_2_1_1"><i><font color="Green" title="E100">A20</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c17">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:105_2_1_1"/><i><font color="Green" title="E102">A22</font></i>: 
<font color="Maroon" title="c13">Q1</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> <font color="Maroon" title="c17">n</font>
 <b>by </b><i><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">Th43</a></i>;<br/><a NAME="E11:105_2_1_1"/><i><font color="Green" title="E103">A23</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c13">Q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Q1</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E10:105_2_1_1"><i><font color="Green" title="E102">A22</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E12:105_2_1_1"/><i><font color="Green" title="E104">A24</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M'</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><a NAME="E13:105_2_1_1"/><b>then </b><i><font color="Green" title="E105">A25</font></i>: 
( <font color="Maroon" title="c13">Q1</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="c6">M'</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M'</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E5:105"><i><font color="Green" title="E94">A4</font></i></a>, <a class="txt" href="matrix13.html#E8:105"><i><font color="Green" title="E96">A16</font></i></a>, <a class="ref" href="matrix13.html#T67" target="_self" title="MATRIX13:th.67">Th67</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:105_2_1_1"/><b>then </b>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span>,<font color="Maroon" title="c16">x</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="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c13">Q1</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E6:105_2_1_1"><i><font color="Green" title="E100">A20</font></i></a>, <a class="txt" href="matrix13.html#E7:105_2_1_1"><i><font color="Green" title="E101">A21</font></i></a>, <a class="ref" href="matrix13.html#T47" target="_self" title="MATRIX13:th.47">Th47</a></i>;<br/><a NAME="E15:105_2_1_1"/><b>then </b><i><font color="Green" title="E106">A26</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span>,<font color="Maroon" title="c16">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c13">Q1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E11:105_2_1_1"><i><font color="Green" title="E103">A23</font></i></a>, <a class="txt" href="matrix13.html#E13:105_2_1_1"><i><font color="Green" title="E105">A25</font></i></a>, <a class="ref" href="matrix13.html#T87" target="_self" title="MATRIX13:th.87">Th87</a>, <a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>;<br/> <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span>,<font color="Maroon" title="c16">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span>,<font color="Maroon" title="c16">x</font></span>)</span></span>)</span> = 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span>,<font color="Maroon" title="c16">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span>,<font color="Maroon" title="c16">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E8:105"><i><font color="Green" title="E96">A16</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></i>
<br/>.= 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span>,<font color="Maroon" title="c16">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span>,<font color="Maroon" title="c16">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E8:105"><i><font color="Green" title="E96">A16</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></i>
<br/>.= 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font>
<b>by </b><i><a class="txt" href="matrix13.html#E5:105"><i><font color="Green" title="E94">A4</font></i></a>, <a class="txt" href="matrix13.html#E8:105"><i><font color="Green" title="E96">A16</font></i></a>, <a class="txt" href="matrix13.html#E6:105_2_1_1"><i><font color="Green" title="E100">A20</font></i></a>, <a class="txt" href="matrix13.html#E12:105_2_1_1"><i><font color="Green" title="E104">A24</font></i></a>, <a class="txt" href="matrix13.html#E15:105_2_1_1"><i><font color="Green" title="E106">A26</font></i></a>, <a class="ref" href="matrix13.html#T59" target="_self" title="MATRIX13:th.59">Th59</a></i>
<br/>.= 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font>
<b>by </b><i><a class="txt" href="matrix13.html#E8:105"><i><font color="Green" title="E96">A16</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></i>
;<br/><a NAME="E17:105_2_1_1"/><b>then </b>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E6:105_2_1_1"><i><font color="Green" title="E100">A20</font></i></a>, <a class="txt" href="matrix13.html#E7:105_2_1_1"><i><font color="Green" title="E101">A21</font></i></a>, <a class="ref" href="matrix11.html#T35" title="MATRIX11:th.35">MATRIX11:35</a></i>;<br/><a NAME="E18:105_2_1_1"/><b>then </b>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">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="c4">K</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E9:105"><i><font color="Green" title="E97">A17</font></i></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>;<br/><b>hence </b><a NAME="E19:105_2_1_1"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c6">M'</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E5:105"><i><font color="Green" title="E94">A4</font></i></a>, <a class="txt" href="matrix13.html#E8:105"><i><font color="Green" title="E96">A16</font></i></a>, <a class="txt" href="matrix13.html#E9:105"><i><font color="Green" title="E97">A17</font></i></a>, <a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_2"><b>suppose </b></a><a NAME="E1:105_2_1_2"/><i><font color="Green" title="E98">A27</font></i>: 
not <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">P1</font>
 ;<br/><div class="add"> <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font> = 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font>
<b>by </b><i><a class="txt" href="matrix13.html#E8:105"><i><font color="Green" title="E96">A16</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></i>
<br/>.= 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font>
<b>by </b><i><a class="txt" href="matrix13.html#E5:105"><i><font color="Green" title="E94">A4</font></i></a>, <a class="txt" href="matrix13.html#E8:105"><i><font color="Green" title="E96">A16</font></i></a>, <a class="txt" href="matrix13.html#E1:105_2_1_2"><i><font color="Green" title="E98">A27</font></i></a>, <a class="ref" href="matrix13.html#T60" target="_self" title="MATRIX13:th.60">Th60</a></i>
<br/>.= 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font>
<b>by </b><i><a class="txt" href="matrix13.html#E8:105"><i><font color="Green" title="E96">A16</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></i>
;<br/><b>hence </b><a NAME="E3:105_2_1_2"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c6">M'</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E5:105"><i><font color="Green" title="E94">A4</font></i></a>, <a class="txt" href="matrix13.html#E8:105"><i><font color="Green" title="E96">A16</font></i></a>, <a class="txt" href="matrix13.html#E9:105"><i><font color="Green" title="E97">A17</font></i></a>, <a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:105"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c6">M'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M'</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E6:105"><i><font color="Green" title="E95">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


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