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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c4">nt</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c5">mt</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">m</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c6">M</font> be   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c3">K</font>;<br/>











<b>set </b><font color="Maroon" title="c7">RM</font> =  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c6">M</font>;<br/>
<b>set </b><font color="Maroon" title="c8">S</font> =  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt</font>,<font color="Maroon" title="c5">mt</font>;<br/>
<b>assume </b><a NAME="E1:96"/><i><font color="Green" title="E86">A1</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">nt</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">mt</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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="c6">M</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">rt1</font>, <font color="Maroon" title="c10">rt2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:96"/><i><font color="Green" title="E87">A2</font></i>: 
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">rt1</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">rt2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">M</font> &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c9">rt1</font>,<font color="Maroon" title="c10">rt2</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="c3">K</font> )
 <b>by </b><i><a class="ref" href="matrix13.html#T76" target="_self" title="MATRIX13:th.76">TT76</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_1_1"/>
(  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c6">M</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1"><b>suppose </b></a><a NAME="E1:96_1_1_1"/>
 <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:96_1_1_1"/>
 <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c6">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt</font>,<font color="Maroon" title="c5">mt</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2"><b>suppose </b></a><a NAME="E1:96_1_1_2"/>
 <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c6">M</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:96_1_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">rt1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">rt2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><a NAME="E3:96_1_1_2"/><b>then </b>
( <font color="Maroon" title="c9">rt1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c10">rt2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><a NAME="E4:96_1_1_2"/><b>then </b><i><font color="Green" title="E88">A3'</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">rt1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">rt2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="ref" href="finseq_1.html#T27" title="FINSEQ_1:th.27">FINSEQ_1:27</a></i>;<br/><a NAME="E5:96_1_1_2"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">rt1</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">rt2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></i>;<br/><a NAME="E6:96_1_1_2"/><b>then </b><i><font color="Green" title="E89">A3</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">rt1</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> <font color="Maroon" title="c4">nt</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">rt2</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> <font color="Maroon" title="c5">mt</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:96"><i><font color="Green" title="E86">A1</font></i></a>, <a class="txt" href="matrix13.html#E3:96"><i><font color="Green" title="E87">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T138" title="ZFMISC_1:th.138">ZFMISC_1:138</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">R1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:96_1_1_2"/><i><font color="Green" title="E90">A4</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c9">rt1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">R1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c4">nt</font> &amp; <font color="Maroon" title="c9">rt1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">nt</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">R1</font> )
 <b>by </b><i><a class="ref" href="matrix13.html#T81" target="_self" title="MATRIX13:th.81">TT81</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">R2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E10:96_1_1_2"/><i><font color="Green" title="E91">A5</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c10">rt2</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">R2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c5">mt</font> &amp; <font color="Maroon" title="c10">rt2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">mt</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">R2</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E6:96_1_1_2"><i><font color="Green" title="E89">A3</font></i></a>, <a class="ref" href="matrix13.html#T81" target="_self" title="MATRIX13:th.81">TT81</a></i>;<br/><a NAME="E11:96_1_1_2"/><i><font color="Green" title="E92">A6</font></i>: 
(  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c9">rt1</font> <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="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c6">M</font></span>)</span> &amp;  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c10">rt2</font> <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="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c6">M</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">R1</font> = <font color="Maroon" title="c11">R1</font>, <font color="Maroon" title="c14">R2</font> = <font color="Maroon" title="c12">R2</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="txt" href="matrix13.html#E8:96_1_1_2"><i><font color="Green" title="E90">A4</font></i></a>, <a class="txt" href="matrix13.html#E10:96_1_1_2"><i><font color="Green" title="E91">A5</font></i></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E13:96_1_1_2"/><i><font color="Green" title="E93">A7</font></i>: 
(  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c4">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c5">mt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a></i>;<br/><a NAME="E14:96_1_1_2"/><b>then </b>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">R1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">R2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="txt" href="matrix13.html#E8:96_1_1_2"><i><font color="Green" title="E90">A4</font></i></a>, <a class="txt" href="matrix13.html#E10:96_1_1_2"><i><font color="Green" title="E91">A5</font></i></a>, <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="c15">R1</font> = <font color="Maroon" title="c13">R1</font>, <font color="Maroon" title="c16">R2</font> = <font color="Maroon" title="c14">R2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E16:96_1_1_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c6">M</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c6">M</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E8:96_1_1_2"><i><font color="Green" title="E90">A4</font></i></a>, <a class="txt" href="matrix13.html#E10:96_1_1_2"><i><font color="Green" title="E91">A5</font></i></a>, <a class="txt" href="matrix13.html#E11:96_1_1_2"><i><font color="Green" title="E92">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">R1</font> = <font color="Maroon" title="c15">R1</font>, <font color="Maroon" title="c18">R2</font> = <font color="Maroon" title="c16">R2</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/><a NAME="E18:96_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">nt</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="matrix13.html#E6:96_1_1_2"><i><font color="Green" title="E89">A3</font></i></a>, <a class="txt" href="matrix13.html#E4:96_1_1_2"><i><font color="Green" title="E88">A3'</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><a NAME="E19:96_1_1_2"/><b>then </b>
(  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c4">nt</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c4">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a>, <a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a></i>;<br/><a NAME="E20:96_1_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt</font>,<font color="Maroon" title="c5">mt</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt</font>,<font color="Maroon" title="c5">mt</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )
 <b>by </b><i><a class="ref" href="matrix13.html#T1" target="_self" title="MATRIX13:th.1">TT16</a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></i>;<br/><a NAME="E21:96_1_1_2"/><b>then </b><i><font color="Green" title="E94">A8</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt</font>,<font color="Maroon" title="c5">mt</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon" title="c19">SR</font> =  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt</font>,<font color="Maroon" title="c5">mt</font></span>)</span>,<font color="Maroon" title="c17">R1</font>,<font color="Maroon" title="c18">R2</font>;<br/><b>set </b><font color="Maroon" title="c20">MR</font> =  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c9">rt1</font>,<font color="Maroon" title="c10">rt2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">i</font>, <font color="Maroon" title="c22">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:96_1_1_2_1"/><i><font color="Green" title="E95">A7</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i</font>,<font color="Maroon" title="c22">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p2">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt</font>,<font color="Maroon" title="c5">mt</font></span>)</span>,<font color="Maroon" title="c17">R1</font>,<font color="Maroon" title="c18">R2</font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c23">I</font> = <font color="Maroon" title="c21">i</font>, <font color="Maroon" title="c24">J</font> = <font color="Maroon" title="c22">j</font>, <font color="Maroon" title="c25">RI</font> = <font color="Maroon" title="c17">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>, <font color="Maroon" title="c26">RJ</font> = <font color="Maroon" title="c18">R2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:96_1_1_2_1"/>
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p2">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt</font>,<font color="Maroon" title="c5">mt</font></span>)</span>,<font color="Maroon" title="c17">R1</font>,<font color="Maroon" title="c18">R2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c17">R1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c18">R2</font></span>)</span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 <b>by </b><i><a class="txt" href="matrix13.html#E8:96_1_1_2"><i><font color="Green" title="E90">A4</font></i></a>, <a class="txt" href="matrix13.html#E10:96_1_1_2"><i><font color="Green" title="E91">A5</font></i></a>, <a class="txt" href="matrix13.html#E11:96_1_1_2"><i><font color="Green" title="E92">A6</font></i></a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/><a NAME="E4:96_1_1_2_1"/><b>then </b><i><font color="Green" title="E96">A9</font></i>: 
( <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c17">R1</font> &amp; <font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c18">R2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i</font>,<font color="Maroon" title="c22">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c9">rt1</font>,<font color="Maroon" title="c10">rt2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:96_1_1_2_1"><i><font color="Green" title="E95">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a>, <a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></i>;<br/><a NAME="E5:96_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c17">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">R1</font> &amp; <font color="Maroon" title="c18">R2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">R2</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:96_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c17">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c4">nt</font> &amp; <font color="Maroon" title="c18">R2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c5">mt</font> &amp;  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c4">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c5">mt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E8:96_1_1_2"><i><font color="Green" title="E90">A4</font></i></a>, <a class="txt" href="matrix13.html#E10:96_1_1_2"><i><font color="Green" title="E91">A5</font></i></a>, <a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a></i>;<br/><a NAME="E7:96_1_1_2_1"/><b>then </b><i><font color="Green" title="E97">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">I</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">R2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">J</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt</font>,<font color="Maroon" title="c5">mt</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E21:96_1_1_2"><i><font color="Green" title="E94">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c9">rt1</font>,<font color="Maroon" title="c10">rt2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i</font>,<font color="Maroon" title="c22">j</font> = 
<font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">rt1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">rt2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">J</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E4:96_1_1_2_1"><i><font color="Green" title="E96">A9</font></i></a>, <a class="ref" href="matrix13.html#D1" target="_self" title="MATRIX13:def.1">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">RI</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">rt2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">J</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E8:96_1_1_2"><i><font color="Green" title="E90">A4</font></i></a>, <a class="txt" href="matrix13.html#E4:96_1_1_2_1"><i><font color="Green" title="E96">A9</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">RI</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">mt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">RJ</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E10:96_1_1_2"><i><font color="Green" title="E91">A5</font></i></a>, <a class="txt" href="matrix13.html#E4:96_1_1_2_1"><i><font color="Green" title="E96">A9</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt</font>,<font color="Maroon" title="c5">mt</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">R2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">J</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E7:96_1_1_2_1"><i><font color="Green" title="E97">A10</font></i></a>, <a class="ref" href="matrix13.html#D1" target="_self" title="MATRIX13:def.1">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <span class="p2">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt</font>,<font color="Maroon" title="c5">mt</font></span>)</span>,<font color="Maroon" title="c17">R1</font>,<font color="Maroon" title="c18">R2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">i</font>,<font color="Maroon" title="c22">j</font>
<b>by </b><i><a class="txt" href="matrix13.html#E1:96_1_1_2_1"><i><font color="Green" title="E95">A7</font></i></a>, <a class="ref" href="matrix13.html#D1" target="_self" title="MATRIX13:def.1">Def2</a></i>
;<br/></div><b>end;</b></div><a NAME="E23:96_1_1_2"/><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#K1" title="MATRIX13:func.1">Segm</a> <span class="p2">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt</font>,<font color="Maroon" title="c5">mt</font></span>)</span>,<font color="Maroon" title="c17">R1</font>,<font color="Maroon" title="c18">R2</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="c3">K</font> &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"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">R1</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">R2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt</font>,<font color="Maroon" title="c5">mt</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E3:96"><i><font color="Green" title="E87">A2</font></i></a>, <a class="txt" href="matrix13.html#E8:96_1_1_2"><i><font color="Green" title="E90">A4</font></i></a>, <a class="txt" href="matrix13.html#E10:96_1_1_2"><i><font color="Green" title="E91">A5</font></i></a>, <a class="txt" href="matrix13.html#E13:96_1_1_2"><i><font color="Green" title="E93">A7</font></i></a>, <a class="txt" href="matrix13.html#E21:96_1_1_2"><i><font color="Green" title="E94">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a>, <a class="ref" href="matrix_1.html#T28" title="MATRIX_1:th.28">MATRIX_1:28</a></i>;<br/><b>hence </b><a NAME="E24:96_1_1_2"/>
 <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c6">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt</font>,<font color="Maroon" title="c5">mt</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix13.html#T76" target="_self" title="MATRIX13:th.76">TT76</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E5:96"/><b>then </b>
(  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c6">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt</font>,<font color="Maroon" title="c5">mt</font></span>)</span> &amp;  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c6">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt</font>,<font color="Maroon" title="c5">mt</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E1:96"><i><font color="Green" title="E86">A1</font></i></a>, <a class="ref" href="matrix13.html#T78" target="_self" title="MATRIX13:th.78">TT78</a></i>;<br/>
<b>hence </b><a NAME="E6:96"/>
 <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt</font>,<font color="Maroon" title="c5">mt</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div>
