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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">j</font>, <font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c5">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">K</font>;<br/><b>let </b><font color="Maroon" title="c7">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="c5">K</font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:108"/><i><font color="Green" title="E94">A1</font></i>: 
<font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M'</font></span>)</span>
 <b>and </b><br/><a NAME="E2:108"/><i><font color="Green" title="E95">A2</font></i>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font> implies <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c5">K</font></span>)</span> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1"/>
(  not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M'</font></span>)</span> or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M'</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1"><b>suppose </b></a><a NAME="E1:108_1_1"/>
not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M'</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:108_1_1"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="matrix13.html#T40" target="_self" title="MATRIX13:th.40">Th40</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2"><b>suppose </b></a><a NAME="E1:108_1_2"/><i><font color="Green" title="E96">A3</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M'</font></span>)</span>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c8">Li</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>;<br/><b>set </b><font color="Maroon" title="c9">Lj</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font>;<br/><b>set </b><font color="Maroon" title="c10">R</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c11">W</font> =  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">M'</font>;<br/><b>set </b><font color="Maroon" title="c12">KK</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">K</font>;<br/><a NAME="E2:108_1_2"/><i><font color="Green" title="E97">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></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="c7">M'</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n'</font> &amp;  <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="c7">M'</font>,<font color="Maroon" title="c4">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="c7">M'</font> &amp;  <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="c7">M'</font>,<font color="Maroon" title="c3">j</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="c7">M'</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></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="c7">M'</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/><a NAME="E3:108_1_2"/><b>then </b><i><font color="Green" title="E98">A5</font></i>: 
(  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</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="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></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="c7">M'</font>,<font color="Maroon" title="c4">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="c7">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="c7">M'</font>,<font color="Maroon" title="c3">j</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="c7">M'</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></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="c7">M'</font></span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">M'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:108_1_2"><i><font color="Green" title="E96">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix11.html#T28" title="MATRIX11:th.28">MATRIX11:28</a>, <a class="ref" href="matrix11.html#D3" title="MATRIX11:def.3">MATRIX11:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">Li'</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c14">Lj'</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font>, <font color="Maroon" title="c15">LL</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">K</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/><i><font color="Green" title="E99">A6</font></i>:  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> = 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c13">Li'</font>
<b>by </b><i><a class="txt" href="matrix13.html#E2:108_1_2"><i><font color="Green" title="E97">A4</font></i></a>, <a class="txt" href="matrix13.html#E3:108_1_2"><i><font color="Green" title="E98">A5</font></i></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></i>
<br/>.= 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c15">LL</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c13">Li'</font>
<b>by </b><i><a class="txt" href="matrix13.html#E2:108_1_2"><i><font color="Green" title="E97">A4</font></i></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></i>
<br/>.= 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c13">Li'</font>
<b>by </b><i><a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></i>
<br/>.= 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E2:108_1_2"><i><font color="Green" title="E97">A4</font></i></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></i>
<br/>.= 
<font color="Maroon" title="c7">M'</font>
<b>by </b><i><a class="ref" href="matrix11.html#T30" title="MATRIX11:th.30">MATRIX11:30</a></i>
;<br/><i><font color="Green" title="E100">A7</font></i>:  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span> = 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c14">Lj'</font>
<b>by </b><i><a class="txt" href="matrix13.html#E2:108_1_2"><i><font color="Green" title="E97">A4</font></i></a>, <a class="txt" href="matrix13.html#E3:108_1_2"><i><font color="Green" title="E98">A5</font></i></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></i>
<br/>.= 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c15">LL</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c14">Lj'</font>
<b>by </b><i><a class="txt" href="matrix13.html#E2:108_1_2"><i><font color="Green" title="E97">A4</font></i></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></i>
<br/>.= 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c14">Lj'</font>
<b>by </b><i><a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></i>
<br/>.= 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E2:108_1_2"><i><font color="Green" title="E97">A4</font></i></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></i>
;<br/><b>consider </b><font color="Maroon" title="c16">P</font>, <font color="Maroon" title="c17">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="E8:108_1_2"/><i><font color="Green" title="E101">A8</font></i>: 
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">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="c17">Q</font> )
 <b>and </b><br/><a NAME="E9:108_1_2"/><i><font color="Green" title="E102">A9</font></i>: 
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">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="c5">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="E10:108_1_2"/><i><font color="Green" title="E103">A10</font></i>: 
(  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font> &amp;  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E8:108_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></i>;<br/><a NAME="E11:108_1_2"/><i><font color="Green" title="E104">A11</font></i>: 
 <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c7">M'</font>
 <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="E105">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_2_3_1"/>
(  not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P</font> or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:108_1_2_3_1_1"/>
not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P</font>
 ;<br/><div class="add"><a NAME="E2:108_1_2_3_1_1"/><b>then </b>
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E8:108_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="matrix13.html#E11:108_1_2"><i><font color="Green" title="E104">A11</font></i></a>, <a class="ref" href="matrix13.html#T60" target="_self" title="MATRIX13:th.60">Th60</a></i>;<br/><b>hence </b><a NAME="E3:108_1_2_3_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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M'</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E8:108_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="matrix13.html#E9:108_1_2"><i><font color="Green" title="E102">A9</font></i></a>, <a class="txt" href="matrix13.html#E10:108_1_2"><i><font color="Green" title="E103">A10</font></i></a>, <a class="txt" href="matrix13.html#E11:108_1_2"><i><font color="Green" title="E104">A11</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="108_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:108_1_2_3_1_2"/><i><font color="Green" title="E106">A13</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c18">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:108_1_2_3_1_2"/><i><font color="Green" title="E107">A14</font></i>: 
<font color="Maroon" title="c16">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="c18">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="E4:108_1_2_3_1_2"/>
 <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="c16">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">P</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:108_1_2_3_1_2"><i><font color="Green" title="E107">A14</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="c19">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:108_1_2_3_1_2"/><i><font color="Green" title="E108">A15</font></i>: 
( <font color="Maroon" title="c19">m</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="c16">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="c16">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:108_1_2_3_1_2"><i><font color="Green" title="E106">A13</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="E7:108_1_2_3_1_2"/><i><font color="Green" title="E109">A16</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="c16">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="c16">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:108_1_2_3_1_2"><i><font color="Green" title="E107">A14</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="c20">m</font> = <font color="Maroon" title="c19">m</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#E6:108_1_2_3_1_2"><i><font color="Green" title="E108">A15</font></i></a></i>;<br/><a NAME="E9:108_1_2_3_1_2"/><i><font color="Green" title="E110">A17</font></i>: 
( <font color="Maroon" title="c17">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="c7">M'</font></span>)</span> &amp; <font color="Maroon" title="c17">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> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E8:108_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="matrix13.html#E11:108_1_2"><i><font color="Green" title="E104">A11</font></i></a>, <a class="ref" href="matrix13.html#T67" target="_self" title="MATRIX13:th.67">Th67</a></i>;<br/><a NAME="E10:108_1_2_3_1_2"/><b>then </b><i><font color="Green" title="E111">A18</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="c17">Q</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">M'</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E11:108_1_2_3_1_2"/><b>then </b><i><font color="Green" title="E112">A19</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c17">Q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c17">Q</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="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c17">Q</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="c17">Q</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E3:108_1_2"><i><font color="Green" title="E98">A5</font></i></a>, <a class="txt" href="matrix13.html#E9:108_1_2_3_1_2"><i><font color="Green" title="E110">A17</font></i></a>, <a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c21">LjQ</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</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="c17">Q</font></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E13:108_1_2_3_1_2"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">LjQ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">K</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></i>;<br/><a NAME="E14:108_1_2_3_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">LjQ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">K</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c22">LjQ</font> = <font color="Maroon" title="c21">LjQ</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c5">K</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>set </b><font color="Maroon" title="c23">SR</font> =  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font>;<br/><b>set </b><font color="Maroon" title="c24">SM</font> =  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font>;<br/><a NAME="E16:108_1_2_3_1_2"/><i><font color="Green" title="E113">A20</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">LjQ</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="c16">P</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span> <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="c16">P</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">LjQ</font> <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"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c22">LjQ</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E8:108_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="matrix13.html#E11:108_1_2_3_1_2"><i><font color="Green" title="E112">A19</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/><a NAME="E17:108_1_2_3_1_2"/><i><font color="Green" title="E114">A21</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> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</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> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span> <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="c16">P</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">P</font> <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="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><a NAME="E18:108_1_2_3_1_2"/><i><font color="Green" title="E115">A22</font></i>: 
(  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</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> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c4">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="c17">Q</font></span>)</span> &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</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="c7">M'</font>,<font color="Maroon" title="c4">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="c17">Q</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E10:108_1_2"><i><font color="Green" title="E103">A10</font></i></a>, <a class="txt" href="matrix13.html#E6:108_1_2_3_1_2"><i><font color="Green" title="E108">A15</font></i></a>, <a class="txt" href="matrix13.html#E7:108_1_2_3_1_2"><i><font color="Green" title="E109">A16</font></i></a>, <a class="txt" href="matrix13.html#E9:108_1_2_3_1_2"><i><font color="Green" title="E110">A17</font></i></a>, <a class="ref" href="matrix13.html#T47" target="_self" title="MATRIX13:th.47">Th47</a></i>;<br/><a NAME="E19:108_1_2_3_1_2"/><i><font color="Green" title="E116">A23</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c22">LjQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</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="c17">Q</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:108_1_2"><i><font color="Green" title="E98">A5</font></i></a>, <a class="txt" href="matrix13.html#E10:108_1_2_3_1_2"><i><font color="Green" title="E111">A18</font></i></a>, <a class="ref" href="matrix13.html#T87" target="_self" title="MATRIX13:th.87">Th87</a></i>;<br/><a NAME="E20:108_1_2_3_1_2"/><i><font color="Green" title="E117">A24</font></i>: 
(  <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> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</font>,<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#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font> &amp;  <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> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</font>,<font color="Maroon" title="c22">LjQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font> &amp;  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E2:108_1_2"><i><font color="Green" title="E97">A4</font></i></a>, <a class="txt" href="matrix13.html#E3:108_1_2"><i><font color="Green" title="E98">A5</font></i></a>, <a class="txt" href="matrix13.html#E5:108_1_2"><i><font color="Green" title="E99">A6</font></i></a>, <a class="txt" href="matrix13.html#E6:108_1_2"><i><font color="Green" title="E100">A7</font></i></a>, <a class="txt" href="matrix13.html#E8:108_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="matrix13.html#E10:108_1_2"><i><font color="Green" title="E103">A10</font></i></a>, <a class="txt" href="matrix13.html#E6:108_1_2_3_1_2"><i><font color="Green" title="E108">A15</font></i></a>, <a class="txt" href="matrix13.html#E18:108_1_2_3_1_2"><i><font color="Green" title="E115">A22</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font> = 
 <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> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</font>,<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#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix11.html#T30" title="MATRIX11:th.30">MATRIX11:30</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> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</font>,<span class="p1">(<span class="default"><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="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c22">LjQ</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E3:108_1_2"><i><font color="Green" title="E98">A5</font></i></a>, <a class="txt" href="matrix13.html#E10:108_1_2_3_1_2"><i><font color="Green" title="E111">A18</font></i></a>, <a class="txt" href="matrix13.html#E18:108_1_2_3_1_2"><i><font color="Green" title="E115">A22</font></i></a>, <a class="txt" href="matrix13.html#E19:108_1_2_3_1_2"><i><font color="Green" title="E116">A23</font></i></a>, <a class="ref" href="matrix13.html#T88" target="_self" title="MATRIX13:th.88">Th88</a></i>
;<br/><b>then </b><i><font color="Green" title="E118">A25</font></i>:  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span> = 
<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="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c22">LjQ</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E6:108_1_2_3_1_2"><i><font color="Green" title="E108">A15</font></i></a>, <a class="txt" href="matrix13.html#E7:108_1_2_3_1_2"><i><font color="Green" title="E109">A16</font></i></a>, <a class="txt" href="matrix13.html#E16:108_1_2_3_1_2"><i><font color="Green" title="E113">A20</font></i></a>, <a class="txt" href="matrix13.html#E17:108_1_2_3_1_2"><i><font color="Green" title="E114">A21</font></i></a>, <a class="ref" href="matrix11.html#T36" title="MATRIX11:th.36">MATRIX11:36</a></i>
<br/>.= 
<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="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p4">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p5">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</font>,<font color="Maroon" title="c22">LjQ</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E6:108_1_2_3_1_2"><i><font color="Green" title="E108">A15</font></i></a>, <a class="txt" href="matrix13.html#E7:108_1_2_3_1_2"><i><font color="Green" title="E109">A16</font></i></a>, <a class="txt" href="matrix13.html#E16:108_1_2_3_1_2"><i><font color="Green" title="E113">A20</font></i></a>, <a class="ref" href="matrix11.html#T34" title="MATRIX11:th.34">MATRIX11:34</a></i>
<br/>.= 
<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="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E8:108_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="matrix13.html#E20:108_1_2_3_1_2"><i><font color="Green" title="E117">A24</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_3_1_2_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_2_3_1_2_3"/>
(  <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="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">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="c5">K</font> or  <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="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">K</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_3_1_2_3_1"><b>suppose </b></a><a NAME="E1:108_1_2_3_1_2_3_1"/>
 <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="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">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="c5">K</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:108_1_2_3_1_2_3_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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M'</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E8:108_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="matrix13.html#E9:108_1_2"><i><font color="Green" title="E102">A9</font></i></a>, <a class="txt" href="matrix13.html#E11:108_1_2"><i><font color="Green" title="E104">A11</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="108_1_2_3_1_2_3_2"><b>suppose </b></a><a NAME="E1:108_1_2_3_1_2_3_2"/><i><font color="Green" title="E119">A26</font></i>: 
 <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="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">K</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c25">j0</font> = <font color="Maroon" title="c3">j</font> as  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a>  <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#E1:108"><i><font color="Green" title="E94">A1</font></i></a>, <a class="ref" href="pscomp_1.html#D1" title="PSCOMP_1:def.1">PSCOMP_1:def 1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_3_1_2_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_2_3_1_2_3_2_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font> or ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P</font> ) or ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> &amp; not <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_3_1_2_3_2_1_1"><b>suppose </b></a><a NAME="E1:108_1_2_3_1_2_3_2_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font>
 ;<br/><div class="add"><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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span> = 
<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="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E22:108_1_2_3_1_2"><i><font color="Green" title="E118">A25</font></i></a>, <a class="ref" href="matrix11.html#T30" title="MATRIX11:th.30">MATRIX11:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a></i>
;<br/><b>hence </b><a NAME="E3:108_1_2_3_1_2_3_2_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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M'</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E9:108_1_2"><i><font color="Green" title="E102">A9</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2_3_1_2_3_2"><i><font color="Green" title="E119">A26</font></i></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_3_1_2_3_2_1_2"><b>suppose </b></a><a NAME="E1:108_1_2_3_1_2_3_2_1_2"/><i><font color="Green" title="E120">A27</font></i>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P</font> )
 ;<br/><div class="add"><a NAME="E2:108_1_2_3_1_2_3_2_1_2"/>
 <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="c16">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">P</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:108_1_2_3_1_2"><i><font color="Green" title="E107">A14</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="c26">l</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:108_1_2_3_1_2_3_2_1_2"/><i><font color="Green" title="E121">A28</font></i>: 
( <font color="Maroon" title="c26">l</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="c16">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="c16">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j0</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:108_1_2_3_1_2_3_2_1_2"><i><font color="Green" title="E120">A27</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">l</font> = <font color="Maroon" title="c26">l</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#E4:108_1_2_3_1_2_3_2_1_2"><i><font color="Green" title="E121">A28</font></i></a></i>;<br/><a NAME="E6:108_1_2_3_1_2_3_2_1_2"/>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c27">l</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="c7">M'</font>,<font color="Maroon" title="c25">j0</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="c17">Q</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E10:108_1_2"><i><font color="Green" title="E103">A10</font></i></a>, <a class="txt" href="matrix13.html#E7:108_1_2_3_1_2"><i><font color="Green" title="E109">A16</font></i></a>, <a class="txt" href="matrix13.html#E9:108_1_2_3_1_2"><i><font color="Green" title="E110">A17</font></i></a>, <a class="txt" href="matrix13.html#E4:108_1_2_3_1_2_3_2_1_2"><i><font color="Green" title="E121">A28</font></i></a>, <a class="ref" href="matrix13.html#T47" target="_self" title="MATRIX13:th.47">Th47</a></i>;<br/><a NAME="E7:108_1_2_3_1_2_3_2_1_2"/><b>then </b><i><font color="Green" title="E122">A29</font></i>: 
 <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="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</font>,<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#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c27">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E2:108_1_2"><i><font color="Green" title="E97">A4</font></i></a>, <a class="txt" href="matrix13.html#E8:108_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="matrix13.html#E10:108_1_2"><i><font color="Green" title="E103">A10</font></i></a>, <a class="txt" href="matrix13.html#E11:108_1_2"><i><font color="Green" title="E104">A11</font></i></a>, <a class="txt" href="matrix13.html#E6:108_1_2_3_1_2"><i><font color="Green" title="E108">A15</font></i></a>, <a class="txt" href="matrix13.html#E20:108_1_2_3_1_2"><i><font color="Green" title="E117">A24</font></i></a>, <a class="ref" href="matrix13.html#T59" target="_self" title="MATRIX13:th.59">Th59</a></i>;<br/><a NAME="E8:108_1_2_3_1_2_3_2_1_2"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</font>,<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="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c27">l</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E6:108_1_2_3_1_2"><i><font color="Green" title="E108">A15</font></i></a>, <a class="txt" href="matrix13.html#E7:108_1_2_3_1_2"><i><font color="Green" title="E109">A16</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2_3_1_2_3_2_1_2"><i><font color="Green" title="E120">A27</font></i></a>, <a class="txt" href="matrix13.html#E4:108_1_2_3_1_2_3_2_1_2"><i><font color="Green" title="E121">A28</font></i></a>, <a class="ref" href="matrix11.html#T51" title="MATRIX11:th.51">MATRIX11:51</a></i>;<br/><b>then </b> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">K</font> = 
<font color="Maroon" title="c6">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c20">m</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span>,<font color="Maroon" title="c27">l</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>
<br/>.= 
<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="c7">M'</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E1:108_1_2_3_1_2_3_2"><i><font color="Green" title="E119">A26</font></i></a>, <a class="txt" href="matrix13.html#E7:108_1_2_3_1_2_3_2_1_2"><i><font color="Green" title="E122">A29</font></i></a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
;<br/><b>hence </b><a NAME="E10:108_1_2_3_1_2_3_2_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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M'</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E9:108_1_2"><i><font color="Green" title="E102">A9</font></i></a>, <a class="txt" href="matrix13.html#E22:108_1_2_3_1_2"><i><font color="Green" title="E118">A25</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_3_1_2_3_2_1_3"><b>suppose </b></a><a NAME="E1:108_1_2_3_1_2_3_2_1_3"/><i><font color="Green" title="E120">A30</font></i>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> &amp; not <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P</font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c26">Pij</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c16">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c25">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>;<br/><a NAME="E2:108_1_2_3_1_2_3_2_1_3"/>
<font color="Maroon" title="c6">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span></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="c5">K</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E9:108_1_2"><i><font color="Green" title="E102">A9</font></i></a>, <a class="txt" href="matrix13.html#E22:108_1_2_3_1_2"><i><font color="Green" title="E118">A25</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2_3_1_2_3_2"><i><font color="Green" title="E119">A26</font></i></a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>;<br/><a NAME="E3:108_1_2_3_1_2_3_2_1_3"/><b>then </b><i><font color="Green" title="E121">A31</font></i>: 
 <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">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="c5">K</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>;<br/><a NAME="E4:108_1_2_3_1_2_3_2_1_3"/>
(  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">M'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c25">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c17">Q</font></span>)</span> or  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</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="c7">M'</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c25">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c17">Q</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:108"><i><font color="Green" title="E94">A1</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2"><i><font color="Green" title="E96">A3</font></i></a>, <a class="txt" href="matrix13.html#E2:108_1_2"><i><font color="Green" title="E97">A4</font></i></a>, <a class="txt" href="matrix13.html#E8:108_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="matrix13.html#E11:108_1_2"><i><font color="Green" title="E104">A11</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2_3_1_2"><i><font color="Green" title="E106">A13</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2_3_1_2_3_2_1_3"><i><font color="Green" title="E120">A30</font></i></a>, <a class="ref" href="matrix13.html#T68" target="_self" title="MATRIX13:th.68">Th68</a></i>;<br/><a NAME="E5:108_1_2_3_1_2_3_2_1_3"/><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="c7">M'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c25">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c17">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="c5">K</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c25">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c25">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c17">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="c7">M'</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:108"><i><font color="Green" title="E94">A1</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2"><i><font color="Green" title="E96">A3</font></i></a>, <a class="txt" href="matrix13.html#E2:108_1_2"><i><font color="Green" title="E97">A4</font></i></a>, <a class="txt" href="matrix13.html#E8:108_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="matrix13.html#E11:108_1_2"><i><font color="Green" title="E104">A11</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2_3_1_2"><i><font color="Green" title="E106">A13</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2_3_1_2_3_2_1_3"><i><font color="Green" title="E120">A30</font></i></a>, <a class="txt" href="matrix13.html#E3:108_1_2_3_1_2_3_2_1_3"><i><font color="Green" title="E121">A31</font></i></a>, <a class="ref" href="matrix13.html#T68" target="_self" title="MATRIX13:th.68">Th68</a>, <a class="ref" href="vectsp_2.html#T34" title="VECTSP_2:th.34">VECTSP_2:34</a></i>;<br/><b>hence </b><a NAME="E6:108_1_2_3_1_2_3_2_1_3"/>
 <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M'</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E8:108_1_2"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="matrix13.html#E9:108_1_2"><i><font color="Green" title="E102">A9</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></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c18">P1</font>, <font color="Maroon" title="c19">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="E14:108_1_2"/><i><font color="Green" title="E106">A32</font></i>: 
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">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> <font color="Maroon" title="c7">M'</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">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="c19">Q1</font> )
 <b>and </b><br/><a NAME="E15:108_1_2"/><i><font color="Green" title="E107">A33</font></i>: 
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">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> <font color="Maroon" title="c7">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="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">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="c5">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="E16:108_1_2"/><i><font color="Green" title="E108">A34</font></i>: 
(  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font> &amp;  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E14:108_1_2"><i><font color="Green" title="E106">A32</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_2_4_1"/>
(  not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">P1</font> or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">P1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_4_1_1"><b>suppose </b></a><a NAME="E1:108_1_2_4_1_1"/>
not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">P1</font>
 ;<br/><div class="add"><a NAME="E2:108_1_2_4_1_1"/><b>then </b>
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E14:108_1_2"><i><font color="Green" title="E106">A32</font></i></a>, <a class="ref" href="matrix13.html#T60" target="_self" title="MATRIX13:th.60">Th60</a></i>;<br/><b>hence </b><a NAME="E3:108_1_2_4_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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></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="c7">M'</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E11:108_1_2"><i><font color="Green" title="E104">A11</font></i></a>, <a class="txt" href="matrix13.html#E14:108_1_2"><i><font color="Green" title="E106">A32</font></i></a>, <a class="txt" href="matrix13.html#E15:108_1_2"><i><font color="Green" title="E107">A33</font></i></a>, <a class="txt" href="matrix13.html#E16:108_1_2"><i><font color="Green" title="E108">A34</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="108_1_2_4_1_2"><b>suppose </b></a><a NAME="E1:108_1_2_4_1_2"/><i><font color="Green" title="E109">A35</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">P1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c20">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:108_1_2_4_1_2"/><i><font color="Green" title="E110">A36</font></i>: 
<font color="Maroon" title="c18">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="c20">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="E4:108_1_2_4_1_2"/>
 <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="c18">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">P1</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:108_1_2_4_1_2"><i><font color="Green" title="E110">A36</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="c21">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:108_1_2_4_1_2"/><i><font color="Green" title="E111">A37</font></i>: 
( <font color="Maroon" title="c21">m</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="c18">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="c18">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:108_1_2_4_1_2"><i><font color="Green" title="E109">A35</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="E7:108_1_2_4_1_2"/><i><font color="Green" title="E112">A38</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="c18">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="c18">P1</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:108_1_2_4_1_2"><i><font color="Green" title="E110">A36</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="c22">m</font> = <font color="Maroon" title="c21">m</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#E6:108_1_2_4_1_2"><i><font color="Green" title="E111">A37</font></i></a></i>;<br/><a NAME="E9:108_1_2_4_1_2"/><i><font color="Green" title="E113">A39</font></i>: 
( <font color="Maroon" title="c19">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="c7">M'</font></span>)</span> &amp; <font color="Maroon" title="c19">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> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E11:108_1_2"><i><font color="Green" title="E104">A11</font></i></a>, <a class="txt" href="matrix13.html#E14:108_1_2"><i><font color="Green" title="E106">A32</font></i></a>, <a class="ref" href="matrix13.html#T67" target="_self" title="MATRIX13:th.67">Th67</a></i>;<br/><a NAME="E10:108_1_2_4_1_2"/><b>then </b><i><font color="Green" title="E114">A40</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="c19">Q1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">M'</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E11:108_1_2_4_1_2"/><b>then </b><i><font color="Green" title="E115">A41</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c19">Q1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c19">Q1</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="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c19">Q1</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="c19">Q1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E3:108_1_2"><i><font color="Green" title="E98">A5</font></i></a>, <a class="txt" href="matrix13.html#E9:108_1_2_4_1_2"><i><font color="Green" title="E113">A39</font></i></a>, <a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c23">LjQ</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</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="c19">Q1</font></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E13:108_1_2_4_1_2"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c23">LjQ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">K</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></i>;<br/><a NAME="E14:108_1_2_4_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c23">LjQ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">K</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c24">LjQ</font> = <font color="Maroon" title="c23">LjQ</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c5">K</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>set </b><font color="Maroon" title="c25">SR</font> =  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font>;<br/><b>set </b><font color="Maroon" title="c26">SM</font> =  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font>;<br/><a NAME="E16:108_1_2_4_1_2"/><i><font color="Green" title="E116">A42</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">LjQ</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="c18">P1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span> <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="c18">P1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">LjQ</font> <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"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c24">LjQ</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E14:108_1_2"><i><font color="Green" title="E106">A32</font></i></a>, <a class="txt" href="matrix13.html#E11:108_1_2_4_1_2"><i><font color="Green" title="E115">A41</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/><a NAME="E17:108_1_2_4_1_2"/><i><font color="Green" title="E117">A43</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> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</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> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span> <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="c18">P1</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">P1</font> <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="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><a NAME="E18:108_1_2_4_1_2"/><i><font color="Green" title="E118">A44</font></i>: 
(  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</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> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c4">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="c19">Q1</font></span>)</span> &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</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="c7">M'</font>,<font color="Maroon" title="c4">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="c19">Q1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E16:108_1_2"><i><font color="Green" title="E108">A34</font></i></a>, <a class="txt" href="matrix13.html#E6:108_1_2_4_1_2"><i><font color="Green" title="E111">A37</font></i></a>, <a class="txt" href="matrix13.html#E7:108_1_2_4_1_2"><i><font color="Green" title="E112">A38</font></i></a>, <a class="txt" href="matrix13.html#E9:108_1_2_4_1_2"><i><font color="Green" title="E113">A39</font></i></a>, <a class="ref" href="matrix13.html#T47" target="_self" title="MATRIX13:th.47">Th47</a></i>;<br/><a NAME="E19:108_1_2_4_1_2"/><i><font color="Green" title="E119">A45</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c24">LjQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</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="c19">Q1</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:108_1_2"><i><font color="Green" title="E98">A5</font></i></a>, <a class="txt" href="matrix13.html#E10:108_1_2_4_1_2"><i><font color="Green" title="E114">A40</font></i></a>, <a class="ref" href="matrix13.html#T87" target="_self" title="MATRIX13:th.87">Th87</a></i>;<br/><a NAME="E20:108_1_2_4_1_2"/><i><font color="Green" title="E120">A46</font></i>: 
(  <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> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</font>,<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#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font> &amp;  <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> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</font>,<font color="Maroon" title="c24">LjQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font> &amp;  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E2:108_1_2"><i><font color="Green" title="E97">A4</font></i></a>, <a class="txt" href="matrix13.html#E3:108_1_2"><i><font color="Green" title="E98">A5</font></i></a>, <a class="txt" href="matrix13.html#E5:108_1_2"><i><font color="Green" title="E99">A6</font></i></a>, <a class="txt" href="matrix13.html#E6:108_1_2"><i><font color="Green" title="E100">A7</font></i></a>, <a class="txt" href="matrix13.html#E11:108_1_2"><i><font color="Green" title="E104">A11</font></i></a>, <a class="txt" href="matrix13.html#E14:108_1_2"><i><font color="Green" title="E106">A32</font></i></a>, <a class="txt" href="matrix13.html#E16:108_1_2"><i><font color="Green" title="E108">A34</font></i></a>, <a class="txt" href="matrix13.html#E6:108_1_2_4_1_2"><i><font color="Green" title="E111">A37</font></i></a>, <a class="txt" href="matrix13.html#E18:108_1_2_4_1_2"><i><font color="Green" title="E118">A44</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font> = 
 <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> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</font>,<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#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix11.html#T30" title="MATRIX11:th.30">MATRIX11:30</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> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</font>,<span class="p1">(<span class="default"><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="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c24">LjQ</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E3:108_1_2"><i><font color="Green" title="E98">A5</font></i></a>, <a class="txt" href="matrix13.html#E10:108_1_2_4_1_2"><i><font color="Green" title="E114">A40</font></i></a>, <a class="txt" href="matrix13.html#E18:108_1_2_4_1_2"><i><font color="Green" title="E118">A44</font></i></a>, <a class="txt" href="matrix13.html#E19:108_1_2_4_1_2"><i><font color="Green" title="E119">A45</font></i></a>, <a class="ref" href="matrix13.html#T88" target="_self" title="MATRIX13:th.88">Th88</a></i>
;<br/><b>then </b><i><font color="Green" title="E121">A47</font></i>:  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span> = 
<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="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c24">LjQ</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E6:108_1_2_4_1_2"><i><font color="Green" title="E111">A37</font></i></a>, <a class="txt" href="matrix13.html#E7:108_1_2_4_1_2"><i><font color="Green" title="E112">A38</font></i></a>, <a class="txt" href="matrix13.html#E16:108_1_2_4_1_2"><i><font color="Green" title="E116">A42</font></i></a>, <a class="txt" href="matrix13.html#E17:108_1_2_4_1_2"><i><font color="Green" title="E117">A43</font></i></a>, <a class="ref" href="matrix11.html#T36" title="MATRIX11:th.36">MATRIX11:36</a></i>
<br/>.= 
<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="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p4">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p5">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</font>,<font color="Maroon" title="c24">LjQ</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E6:108_1_2_4_1_2"><i><font color="Green" title="E111">A37</font></i></a>, <a class="txt" href="matrix13.html#E7:108_1_2_4_1_2"><i><font color="Green" title="E112">A38</font></i></a>, <a class="txt" href="matrix13.html#E16:108_1_2_4_1_2"><i><font color="Green" title="E116">A42</font></i></a>, <a class="ref" href="matrix11.html#T34" title="MATRIX11:th.34">MATRIX11:34</a></i>
<br/>.= 
<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="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E14:108_1_2"><i><font color="Green" title="E106">A32</font></i></a>, <a class="txt" href="matrix13.html#E20:108_1_2_4_1_2"><i><font color="Green" title="E120">A46</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_4_1_2_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_2_4_1_2_3"/>
(  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">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="c5">K</font> or  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">K</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_4_1_2_3_1"><b>suppose </b></a><a NAME="E1:108_1_2_4_1_2_3_1"/>
 <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">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="c5">K</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:108_1_2_4_1_2_3_1"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">M'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E11:108_1_2"><i><font color="Green" title="E104">A11</font></i></a>, <a class="txt" href="matrix13.html#E14:108_1_2"><i><font color="Green" title="E106">A32</font></i></a>, <a class="txt" href="matrix13.html#E15:108_1_2"><i><font color="Green" title="E107">A33</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="108_1_2_4_1_2_3_2"><b>suppose </b></a><a NAME="E1:108_1_2_4_1_2_3_2"/><i><font color="Green" title="E122">A48</font></i>: 
 <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">K</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c27">j0</font> = <font color="Maroon" title="c3">j</font> as  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a>  <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#E1:108"><i><font color="Green" title="E94">A1</font></i></a>, <a class="ref" href="pscomp_1.html#D1" title="PSCOMP_1:def.1">PSCOMP_1:def 1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_4_1_2_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_2_4_1_2_3_2_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font> or ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">P1</font> ) or ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> &amp; not <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">P1</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_4_1_2_3_2_1_1"><b>suppose </b></a><a NAME="E1:108_1_2_4_1_2_3_2_1_1"/><i><font color="Green" title="E123">A49</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font>
 ;<br/><div class="add"><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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span> = 
<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="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E22:108_1_2_4_1_2"><i><font color="Green" title="E121">A47</font></i></a>, <a class="ref" href="matrix11.html#T30" title="MATRIX11:th.30">MATRIX11:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a></i>
;<br/><a NAME="E3:108_1_2_4_1_2_3_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">K</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E15:108_1_2"><i><font color="Green" title="E107">A33</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2_4_1_2_3_2"><i><font color="Green" title="E122">A48</font></i></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>;<br/><b>then </b><font color="Maroon" title="c6">a</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c5">K</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_2.html#T22" title="VECTSP_2:th.22">VECTSP_2:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c5">K</font></span>)</span></span>)</span>

<br/>.= 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c5">K</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
;<br/><b>hence </b><a NAME="E5:108_1_2_4_1_2_3_2_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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></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="c7">M'</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E2:108"><i><font color="Green" title="E95">A2</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2_4_1_2_3_2_1_1"><i><font color="Green" title="E123">A49</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_4_1_2_3_2_1_2"><b>suppose </b></a><a NAME="E1:108_1_2_4_1_2_3_2_1_2"/><i><font color="Green" title="E123">A50</font></i>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">P1</font> )
 ;<br/><div class="add"><a NAME="E2:108_1_2_4_1_2_3_2_1_2"/>
 <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="c18">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">P1</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:108_1_2_4_1_2"><i><font color="Green" title="E110">A36</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="c28">l</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:108_1_2_4_1_2_3_2_1_2"/><i><font color="Green" title="E124">A51</font></i>: 
( <font color="Maroon" title="c28">l</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="c18">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="c18">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">j0</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:108_1_2_4_1_2_3_2_1_2"><i><font color="Green" title="E123">A50</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">l</font> = <font color="Maroon" title="c28">l</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#E4:108_1_2_4_1_2_3_2_1_2"><i><font color="Green" title="E124">A51</font></i></a></i>;<br/><a NAME="E6:108_1_2_4_1_2_3_2_1_2"/>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c29">l</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="c7">M'</font>,<font color="Maroon" title="c27">j0</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="c19">Q1</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E16:108_1_2"><i><font color="Green" title="E108">A34</font></i></a>, <a class="txt" href="matrix13.html#E7:108_1_2_4_1_2"><i><font color="Green" title="E112">A38</font></i></a>, <a class="txt" href="matrix13.html#E9:108_1_2_4_1_2"><i><font color="Green" title="E113">A39</font></i></a>, <a class="txt" href="matrix13.html#E4:108_1_2_4_1_2_3_2_1_2"><i><font color="Green" title="E124">A51</font></i></a>, <a class="ref" href="matrix13.html#T47" target="_self" title="MATRIX13:th.47">Th47</a></i>;<br/><a NAME="E7:108_1_2_4_1_2_3_2_1_2"/><b>then </b><i><font color="Green" title="E125">A52</font></i>: 
 <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="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</font>,<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#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c29">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E2:108_1_2"><i><font color="Green" title="E97">A4</font></i></a>, <a class="txt" href="matrix13.html#E14:108_1_2"><i><font color="Green" title="E106">A32</font></i></a>, <a class="txt" href="matrix13.html#E16:108_1_2"><i><font color="Green" title="E108">A34</font></i></a>, <a class="txt" href="matrix13.html#E6:108_1_2_4_1_2"><i><font color="Green" title="E111">A37</font></i></a>, <a class="txt" href="matrix13.html#E20:108_1_2_4_1_2"><i><font color="Green" title="E120">A46</font></i></a>, <a class="ref" href="matrix13.html#T59" target="_self" title="MATRIX13:th.59">Th59</a></i>;<br/><a NAME="E8:108_1_2_4_1_2_3_2_1_2"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</font>,<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="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c29">l</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E6:108_1_2_4_1_2"><i><font color="Green" title="E111">A37</font></i></a>, <a class="txt" href="matrix13.html#E7:108_1_2_4_1_2"><i><font color="Green" title="E112">A38</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2_4_1_2_3_2_1_2"><i><font color="Green" title="E123">A50</font></i></a>, <a class="txt" href="matrix13.html#E4:108_1_2_4_1_2_3_2_1_2"><i><font color="Green" title="E124">A51</font></i></a>, <a class="ref" href="matrix11.html#T51" title="MATRIX11:th.51">MATRIX11:51</a></i>;<br/><a NAME="E9:108_1_2_4_1_2_3_2_1_2"/><b>then </b>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c22">m</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span>,<font color="Maroon" title="c29">l</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>;<br/><b>hence </b><a NAME="E10:108_1_2_4_1_2_3_2_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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></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="c7">M'</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E15:108_1_2"><i><font color="Green" title="E107">A33</font></i></a>, <a class="txt" href="matrix13.html#E22:108_1_2_4_1_2"><i><font color="Green" title="E121">A47</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2_4_1_2_3_2"><i><font color="Green" title="E122">A48</font></i></a>, <a class="txt" href="matrix13.html#E7:108_1_2_4_1_2_3_2_1_2"><i><font color="Green" title="E125">A52</font></i></a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_4_1_2_3_2_1_3"><b>suppose </b></a><a NAME="E1:108_1_2_4_1_2_3_2_1_3"/><i><font color="Green" title="E123">A53</font></i>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> &amp; not <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">P1</font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c28">Pij</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c27">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>;<br/><a NAME="E2:108_1_2_4_1_2_3_2_1_3"/>
<font color="Maroon" title="c6">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span></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="c5">K</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E15:108_1_2"><i><font color="Green" title="E107">A33</font></i></a>, <a class="txt" href="matrix13.html#E22:108_1_2_4_1_2"><i><font color="Green" title="E121">A47</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2_4_1_2_3_2"><i><font color="Green" title="E122">A48</font></i></a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>;<br/><a NAME="E3:108_1_2_4_1_2_3_2_1_3"/><b>then </b><i><font color="Green" title="E124">A54</font></i>: 
 <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">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="c5">K</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>;<br/><a NAME="E4:108_1_2_4_1_2_3_2_1_3"/>
(  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">M'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c27">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c19">Q1</font></span>)</span> or  <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c18">P1</font>,<font color="Maroon" title="c19">Q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</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="c7">M'</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c18">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c27">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c19">Q1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:108"><i><font color="Green" title="E94">A1</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2"><i><font color="Green" title="E96">A3</font></i></a>, <a class="txt" href="matrix13.html#E2:108_1_2"><i><font color="Green" title="E97">A4</font></i></a>, <a class="txt" href="matrix13.html#E14:108_1_2"><i><font color="Green" title="E106">A32</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2_4_1_2"><i><font color="Green" title="E109">A35</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2_4_1_2_3_2_1_3"><i><font color="Green" title="E123">A53</font></i></a>, <a class="ref" href="matrix13.html#T68" target="_self" title="MATRIX13:th.68">Th68</a></i>;<br/><a NAME="E5:108_1_2_4_1_2_3_2_1_3"/><b>then </b><i><font color="Green" title="E125">A55</font></i>: 
(  <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="c7">M'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c27">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c19">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="c5">K</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c27">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c27">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c19">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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:108"><i><font color="Green" title="E94">A1</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2"><i><font color="Green" title="E96">A3</font></i></a>, <a class="txt" href="matrix13.html#E2:108_1_2"><i><font color="Green" title="E97">A4</font></i></a>, <a class="txt" href="matrix13.html#E11:108_1_2"><i><font color="Green" title="E104">A11</font></i></a>, <a class="txt" href="matrix13.html#E14:108_1_2"><i><font color="Green" title="E106">A32</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2_4_1_2"><i><font color="Green" title="E109">A35</font></i></a>, <a class="txt" href="matrix13.html#E1:108_1_2_4_1_2_3_2_1_3"><i><font color="Green" title="E123">A53</font></i></a>, <a class="txt" href="matrix13.html#E3:108_1_2_4_1_2_3_2_1_3"><i><font color="Green" title="E124">A54</font></i></a>, <a class="ref" href="matrix13.html#T68" target="_self" title="MATRIX13:th.68">Th68</a>, <a class="ref" href="vectsp_2.html#T34" title="VECTSP_2:th.34">VECTSP_2:34</a></i>;<br/><a NAME="E6:108_1_2_4_1_2_3_2_1_3"/>
( not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span> &amp; not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c3">j</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:108_1_2_4_1_2_3_2_1_3"><i><font color="Green" title="E123">A53</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><a NAME="E7:108_1_2_4_1_2_3_2_1_3"/><b>then </b><i><font color="Green" title="E126">A56</font></i>: 
not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c27">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/> <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c7">M'</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c27">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c19">Q1</font> = 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c7">M'</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c27">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c19">Q1</font>
<b>by </b><i><a class="txt" href="matrix13.html#E14:108_1_2"><i><font color="Green" title="E106">A32</font></i></a>, <a class="txt" href="matrix13.html#E5:108_1_2_4_1_2_3_2_1_3"><i><font color="Green" title="E125">A55</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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c27">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c19">Q1</font>
<b>by </b><i><a class="txt" href="matrix13.html#E11:108_1_2"><i><font color="Green" title="E104">A11</font></i></a>, <a class="txt" href="matrix13.html#E5:108_1_2_4_1_2_3_2_1_3"><i><font color="Green" title="E125">A55</font></i></a>, <a class="txt" href="matrix13.html#E7:108_1_2_4_1_2_3_2_1_3"><i><font color="Green" title="E126">A56</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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><font color="Maroon" title="c27">j0</font></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span>)</span>,<font color="Maroon" title="c19">Q1</font>
<b>by </b><i><a class="txt" href="matrix13.html#E14:108_1_2"><i><font color="Green" title="E106">A32</font></i></a>, <a class="txt" href="matrix13.html#E5:108_1_2_4_1_2_3_2_1_3"><i><font color="Green" title="E125">A55</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="E9:108_1_2_4_1_2_3_2_1_3"/>
 <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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></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="c7">M'</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E14:108_1_2"><i><font color="Green" title="E106">A32</font></i></a>, <a class="txt" href="matrix13.html#E15:108_1_2"><i><font color="Green" title="E107">A33</font></i></a>, <a class="txt" href="matrix13.html#E5:108_1_2_4_1_2_3_2_1_3"><i><font color="Green" title="E125">A55</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></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E18:108_1_2"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">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="c7">M'</font>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M'</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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="c7">M'</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E12:108_1_2"><i><font color="Green" title="E105">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
