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<span class="kw">let </span><font color="Maroon" title="c1">n9</font>, <font color="Maroon" title="c2">m9</font>, <font color="Maroon" title="c3">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">K</font><br/> for <font color="Olive" title="b3">M9</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n9</font>,<font color="Maroon" title="c2">m9</font>,<font color="Olive" title="b1">K</font>  st <font color="Olive" title="b2">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="Olive" title="b1">K</font> holds <br/> <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b3">M9</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="Olive" title="b3">M9</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Olive" title="b2">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="Olive" title="b3">M9</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">K</font><br/> for <font color="Olive" title="b2">M9</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n9</font>,<font color="Maroon" title="c2">m9</font>,<font color="Maroon" title="c4">K</font>  st <font color="Olive" title="b1">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> holds <br/> <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b2">M9</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="Olive" title="b2">M9</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">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="Olive" title="b2">M9</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M9</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n9</font>,<font color="Maroon" title="c2">m9</font>,<font color="Maroon" title="c4">K</font>  st <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> holds <br/> <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b1">M9</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="Olive" title="b1">M9</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="Olive" title="b1">M9</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">M9</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n9</font>,<font color="Maroon" title="c2">m9</font>,<font color="Maroon" title="c4">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c6">M9</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">M9</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">M9</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span> )</span><br/>











<span class="kw">set </span><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">M9</font>,<font color="Maroon" title="c3">i</font>;<br/>
<span class="kw">set </span><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">M9</font>,<font color="Maroon" title="c3">i</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">R</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">M9</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">M9</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:105"/><span class="lab"><font color="Green" title="E91">A1</font></span>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">M9</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">M9</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">M9</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></span>;<br/>
<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:105"/><span class="lab"><font color="Green" title="E92">A2</font></span>: 
<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">M9</font>
 <span class="kw">and </span><br/><a NAME="E4:105"/><span class="lab"><font color="Green" title="E93">A3</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E5:105"/><span class="lab"><font color="Green" title="E94">A4</font></span>: 
 <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">M9</font>
 <span class="kw">and </span><br/><a NAME="E6:105"/><span class="lab"><font color="Green" title="E95">A5</font></span>: 
 <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">M9</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:105"/><span class="lab"><font color="Green" title="E96">A6</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c6">M9</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">M9</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">M9</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E97">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></a><a NAME="E1:105_1_1_1"/><span class="lab"><font color="Green" title="E98">A8</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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">M9</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">M9</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">M9</font></span><br/><div class="add"><a NAME="E2:105_1_1_1"/><span class="lab"><font color="Green" title="E99">A9</font></span>: 
 <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">M9</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">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E3:105_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A10</font></span>: 
 <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">M9</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">M9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E4:105_1_1_1"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <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="Olive" title="b1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">Th43</a></span>;<br/><a NAME="E5:105_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A11</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/><a NAME="E6:105_1_1_1"/><span class="lab"><font color="Green" title="E102">A12</font></span>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <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="Olive" title="b1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">Th43</a></span>;<br/><a NAME="E7:105_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A13</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a></span>;<br/><a NAME="E8:105_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:105_1_1_1"><span class="lab"><font color="Green" title="E102">A12</font></span></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:105_1_1_1"/><span class="lab"><font color="Green" title="E104">A14</font></span>: 
<font color="Maroon" title="c12">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>
 <span class="kw">and </span><br/><a NAME="E11:105_1_1_1"/><span class="lab"><font color="Green" title="E105">A15</font></span>: 
<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="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105_1_1_1"><span class="lab"><font color="Green" title="E98">A8</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c12">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:105_1_1_1"><span class="lab"><font color="Green" title="E104">A14</font></span></a></span>;<br/><a NAME="E13:105_1_1_1"/><span class="lab"><font color="Green" title="E106">A16</font></span>: 
<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">M9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:105"><span class="lab"><font color="Green" title="E92">A2</font></span></a>, <a class="txt" href="#E4:105"><span class="lab"><font color="Green" title="E93">A3</font></span></a>, <a class="ref" href="matrix13.html#T67" target="_self" title="MATRIX13:th.67">Th67</a></span>;<br/><a NAME="E14:105_1_1_1"/><span class="kw">then </span>
 <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">M9</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span>,<font color="Maroon" title="c13">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">M9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:105_1_1_1"><span class="lab"><font color="Green" title="E104">A14</font></span></a>, <a class="txt" href="#E11:105_1_1_1"><span class="lab"><font color="Green" title="E105">A15</font></span></a>, <a class="txt" href="#E7:105_1_1_1"><span class="lab"><font color="Green" title="E103">A13</font></span></a>, <a class="ref" href="matrix13.html#T47" target="_self" title="MATRIX13:th.47">Th47</a></span>;<br/><a NAME="E15:105_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A17</font></span>: 
<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">M9</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span>,<font color="Maroon" title="c13">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">M9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:105_1_1_1"><span class="lab"><font color="Green" title="E101">A11</font></span></a>, <a class="txt" href="#E13:105_1_1_1"><span class="lab"><font color="Green" title="E106">A16</font></span></a>, <a class="txt" href="#E3:105_1_1_1"><span class="lab"><font color="Green" title="E100">A10</font></span></a>, <a class="ref" href="matrix13.html#T87" target="_self" title="MATRIX13:th.87">Th87</a></span>;<br/><a NAME="E16:105_1_1_1"/><span class="lab"><font color="Green" title="E108">A18</font></span>: 
 <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">M9</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">M9</font>,<font color="Maroon" title="c3">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>;<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">M9</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span>,<font color="Maroon" title="c13">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">M9</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span>,<font color="Maroon" title="c13">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">M9</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span>,<font color="Maroon" title="c13">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">M9</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span>,<font color="Maroon" title="c13">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:105"><span class="lab"><font color="Green" title="E93">A3</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>
<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">M9</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span>,<font color="Maroon" title="c13">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">M9</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span>,<font color="Maroon" title="c13">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:105"><span class="lab"><font color="Green" title="E93">A3</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>
<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">M9</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">M9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:105"><span class="lab"><font color="Green" title="E92">A2</font></span></a>, <a class="txt" href="#E11:105_1_1_1"><span class="lab"><font color="Green" title="E105">A15</font></span></a>, <a class="txt" href="#E2:105_1_1_1"><span class="lab"><font color="Green" title="E99">A9</font></span></a>, <a class="txt" href="#E15:105_1_1_1"><span class="lab"><font color="Green" title="E107">A17</font></span></a>, <a class="txt" href="#E16:105_1_1_1"><span class="lab"><font color="Green" title="E108">A18</font></span></a>, <a class="ref" href="matrix13.html#T59" target="_self" title="MATRIX13:th.59">Th59</a></span>
<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">M9</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">M9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:105"><span class="lab"><font color="Green" title="E93">A3</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>
;<br/><a NAME="E18:105_1_1_1"/><span class="kw">then </span>
 <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">M9</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">M9</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#K8" title="GROUP_1:func.8">*</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">M9</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:105_1_1_1"><span class="lab"><font color="Green" title="E104">A14</font></span></a>, <a class="txt" href="#E7:105_1_1_1"><span class="lab"><font color="Green" title="E103">A13</font></span></a>, <a class="ref" href="matrix11.html#T35" title="MATRIX11:th.35">MATRIX11:35</a></span>;<br/><a NAME="E19:105_1_1_1"/><span class="kw">then </span>
 <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">M9</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">M9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:105"><span class="lab"><font color="Green" title="E96">A6</font></span></a>, <a class="txt" href="#E6:105"><span class="lab"><font color="Green" title="E95">A5</font></span></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E20:105_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">M9</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">M9</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">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:105"><span class="lab"><font color="Green" title="E92">A2</font></span></a>, <a class="txt" href="#E4:105"><span class="lab"><font color="Green" title="E93">A3</font></span></a>, <a class="txt" href="#E5:105"><span class="lab"><font color="Green" title="E94">A4</font></span></a>, <a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:105_1_1_2"/><span class="lab"><font color="Green" title="E98">A19</font></span>: 
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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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">M9</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">M9</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">M9</font></span><br/><div class="add"> <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c6">M9</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">M9</font>,<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:105"><span class="lab"><font color="Green" title="E93">A3</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>
<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">M9</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">M9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:105"><span class="lab"><font color="Green" title="E92">A2</font></span></a>, <a class="txt" href="#E1:105_1_1_2"><span class="lab"><font color="Green" title="E98">A19</font></span></a>, <a class="ref" href="matrix13.html#T60" target="_self" title="MATRIX13:th.60">Th60</a></span>
<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">M9</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">M9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:105"><span class="lab"><font color="Green" title="E93">A3</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>
;<br/><span class="kw">hence </span><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">M9</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">M9</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">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:105"><span class="lab"><font color="Green" title="E92">A2</font></span></a>, <a class="txt" href="#E4:105"><span class="lab"><font color="Green" title="E93">A3</font></span></a>, <a class="txt" href="#E5:105"><span class="lab"><font color="Green" title="E94">A4</font></span></a>, <a class="txt" href="#E6:105"><span class="lab"><font color="Green" title="E95">A5</font></span></a>, <a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E10:105"/><span class="lab"><font color="Green" title="E98">A20</font></span>: 
<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">M9</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">M9</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:105"/><span class="lab"><font color="Green" title="E99">A21</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E12:105"/><span class="lab"><font color="Green" title="E100">A22</font></span>: 
 <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">M9</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">M9</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E13:105"/><span class="lab"><font color="Green" title="E101">A23</font></span>: 
 <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">M9</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">M9</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></a><a NAME="E1:105_2_1_1"/><span class="lab"><font color="Green" title="E102">A24</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c6">M9</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">M9</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">M9</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:105_2_1_1"/><span class="lab"><font color="Green" title="E103">A25</font></span>: 
 <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">M9</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">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E3:105_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A26</font></span>: 
 <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">M9</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">M9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E4:105_2_1_1"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <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="Olive" title="b1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">Th43</a></span>;<br/><a NAME="E5:105_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A27</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/><a NAME="E6:105_2_1_1"/><span class="lab"><font color="Green" title="E106">A28</font></span>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <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="Olive" title="b1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">Th43</a></span>;<br/><a NAME="E7:105_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A29</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a></span>;<br/><a NAME="E8:105_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:105_2_1_1"><span class="lab"><font color="Green" title="E106">A28</font></span></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:105_2_1_1"/><span class="lab"><font color="Green" title="E108">A30</font></span>: 
<font color="Maroon" title="c14">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>
 <span class="kw">and </span><br/><a NAME="E11:105_2_1_1"/><span class="lab"><font color="Green" title="E109">A31</font></span>: 
<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="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105_2_1_1"><span class="lab"><font color="Green" title="E102">A24</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c14">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:105_2_1_1"><span class="lab"><font color="Green" title="E108">A30</font></span></a></span>;<br/><a NAME="E13:105_2_1_1"/><span class="lab"><font color="Green" title="E110">A32</font></span>: 
<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">M9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="txt" href="#E10:105"><span class="lab"><font color="Green" title="E98">A20</font></span></a>, <a class="txt" href="#E11:105"><span class="lab"><font color="Green" title="E99">A21</font></span></a>, <a class="ref" href="matrix13.html#T67" target="_self" title="MATRIX13:th.67">Th67</a></span>;<br/><a NAME="E14:105_2_1_1"/><span class="kw">then </span>
 <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">M9</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span>,<font color="Maroon" title="c15">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">M9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:105_2_1_1"><span class="lab"><font color="Green" title="E108">A30</font></span></a>, <a class="txt" href="#E11:105_2_1_1"><span class="lab"><font color="Green" title="E109">A31</font></span></a>, <a class="txt" href="#E7:105_2_1_1"><span class="lab"><font color="Green" title="E107">A29</font></span></a>, <a class="ref" href="matrix13.html#T47" target="_self" title="MATRIX13:th.47">Th47</a></span>;<br/><a NAME="E15:105_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A33</font></span>: 
<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">M9</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span>,<font color="Maroon" title="c15">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">M9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:105_2_1_1"><span class="lab"><font color="Green" title="E105">A27</font></span></a>, <a class="txt" href="#E13:105_2_1_1"><span class="lab"><font color="Green" title="E110">A32</font></span></a>, <a class="txt" href="#E3:105_2_1_1"><span class="lab"><font color="Green" title="E104">A26</font></span></a>, <a class="ref" href="matrix13.html#T87" target="_self" title="MATRIX13:th.87">Th87</a></span>;<br/><a NAME="E16:105_2_1_1"/><span class="lab"><font color="Green" title="E112">A34</font></span>: 
 <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">M9</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">M9</font>,<font color="Maroon" title="c3">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>;<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">M9</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span>,<font color="Maroon" title="c15">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">M9</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span>,<font color="Maroon" title="c15">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">M9</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span>,<font color="Maroon" title="c15">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">M9</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span>,<font color="Maroon" title="c15">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:105"><span class="lab"><font color="Green" title="E99">A21</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>
<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">M9</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span>,<font color="Maroon" title="c15">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">M9</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span>,<font color="Maroon" title="c15">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:105"><span class="lab"><font color="Green" title="E99">A21</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>
<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">M9</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">M9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="txt" href="#E10:105"><span class="lab"><font color="Green" title="E98">A20</font></span></a>, <a class="txt" href="#E11:105_2_1_1"><span class="lab"><font color="Green" title="E109">A31</font></span></a>, <a class="txt" href="#E2:105_2_1_1"><span class="lab"><font color="Green" title="E103">A25</font></span></a>, <a class="txt" href="#E15:105_2_1_1"><span class="lab"><font color="Green" title="E111">A33</font></span></a>, <a class="txt" href="#E16:105_2_1_1"><span class="lab"><font color="Green" title="E112">A34</font></span></a>, <a class="ref" href="matrix13.html#T59" target="_self" title="MATRIX13:th.59">Th59</a></span>
<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">M9</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">M9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E11:105"><span class="lab"><font color="Green" title="E99">A21</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>
;<br/><a NAME="E18:105_2_1_1"/><span class="kw">then </span>
 <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">M9</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">M9</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#K8" title="GROUP_1:func.8">*</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">M9</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:105_2_1_1"><span class="lab"><font color="Green" title="E108">A30</font></span></a>, <a class="txt" href="#E7:105_2_1_1"><span class="lab"><font color="Green" title="E107">A29</font></span></a>, <a class="ref" href="matrix11.html#T35" title="MATRIX11:th.35">MATRIX11:35</a></span>;<br/><a NAME="E19:105_2_1_1"/><span class="kw">then </span>
 <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">M9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:105"><span class="lab"><font color="Green" title="E101">A23</font></span></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E20:105_2_1_1"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c6">M9</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">M9</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">M9</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="txt" href="#E10:105"><span class="lab"><font color="Green" title="E98">A20</font></span></a>, <a class="txt" href="#E11:105"><span class="lab"><font color="Green" title="E99">A21</font></span></a>, <a class="txt" href="#E12:105"><span class="lab"><font color="Green" title="E100">A22</font></span></a>, <a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:105_2_1_2"/><span class="lab"><font color="Green" title="E102">A35</font></span>: 
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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c6">M9</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">M9</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">M9</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span></span><br/><div class="add"> <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c6">M9</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">M9</font>,<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:105"><span class="lab"><font color="Green" title="E99">A21</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>
<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">M9</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">M9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="txt" href="#E10:105"><span class="lab"><font color="Green" title="E98">A20</font></span></a>, <a class="txt" href="#E1:105_2_1_2"><span class="lab"><font color="Green" title="E102">A35</font></span></a>, <a class="ref" href="matrix13.html#T60" target="_self" title="MATRIX13:th.60">Th60</a></span>
<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">M9</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">M9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E11:105"><span class="lab"><font color="Green" title="E99">A21</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>
;<br/><span class="kw">hence </span><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">M9</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">M9</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">M9</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="txt" href="#E10:105"><span class="lab"><font color="Green" title="E98">A20</font></span></a>, <a class="txt" href="#E11:105"><span class="lab"><font color="Green" title="E99">A21</font></span></a>, <a class="txt" href="#E12:105"><span class="lab"><font color="Green" title="E100">A22</font></span></a>, <a class="txt" href="#E13:105"><span class="lab"><font color="Green" title="E101">A23</font></span></a>, <a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E15:105"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c6">M9</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">M9</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">M9</font>,<font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:105"><span class="lab"><font color="Green" title="E97">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
