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<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">M</font> be   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">P1</font>, <font color="Maroon" title="c5">Q</font>, <font color="Maroon" title="c6">Q1</font> be  <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> ;<br/>











<b>assume </b><a NAME="E1:94"/><i><font color="Green" title="E84">A1</font></i>: 
( <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P1</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Q1</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c7">S</font> =  <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">Q</font>;<br/>
<b>set </b><font color="Maroon" title="c8">S1</font> =  <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P1</font>,<font color="Maroon" title="c6">Q1</font>;<br/>
<b>consider </b><font color="Maroon" title="c9">P2</font>, <font color="Maroon" title="c10">Q2</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:94"/><i><font color="Green" title="E85">A3</font></i>: 
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c9">P2</font>,<font color="Maroon" title="c10">Q2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">Q2</font> )
 <b>and </b><br/><a NAME="E4:94"/><i><font color="Green" title="E86">A4</font></i>: 
(  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">P2</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#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span> &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">EqSegm</a> <span class="p2">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c9">P2</font>,<font color="Maroon" title="c10">Q2</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="c1">K</font> )
 <b>by </b><i><a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def5</a></i>;<br/>
<a NAME="E5:94"/>
( <font color="Maroon" title="c9">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  iff <font color="Maroon" title="c10">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="matrix13.html#E3:94"><i><font color="Green" title="E85">A3</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">P3</font>, <font color="Maroon" title="c12">Q3</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:94"/><i><font color="Green" title="E87">A5</font></i>: 
( <font color="Maroon" title="c11">P3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c12">Q3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Q</font> )
 <b>and </b><br/><a NAME="E8:94"/>
( <font color="Maroon" title="c11">P3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c9">P2</font> &amp; <font color="Maroon" title="c12">Q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c10">Q2</font> )
 <b>and </b><br/><a NAME="E9:94"/><i><font color="Green" title="E88">A6</font></i>: 
(  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c11">P3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">P2</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c12">Q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">Q2</font> )
 <b>and </b><br/><a NAME="E10:94"/><i><font color="Green" title="E89">A7</font></i>: 
 <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c9">P2</font>,<font color="Maroon" title="c10">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c11">P3</font>,<font color="Maroon" title="c12">Q3</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:94"><i><font color="Green" title="E85">A3</font></i></a>, <a class="ref" href="matrix13.html#T57" target="_self" title="MATRIX13:th.57">TT57</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">P4</font> = <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c4">P1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c11">P3</font>, <font color="Maroon" title="c14">Q4</font> = <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c6">Q1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c12">Q3</font> as  <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>by </b><i><a class="ref" href="matrix13.html#T53" target="_self" title="MATRIX13:th.53">TT53</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">k</font> being    <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> such that </b><br/><a NAME="E13:94"/><i><font color="Green" title="E90">A8</font></i>: 
<font color="Maroon" title="c13">P4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">TT43</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">k</font> being    <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> such that </b><br/><a NAME="E15:94"/><i><font color="Green" title="E91">A9</font></i>: 
<font color="Maroon" title="c14">Q4</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="c16">k</font>
 <b>by </b><i><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">TT43</a></i>;<br/>
<a NAME="E16:94"/>
( <font color="Maroon" title="c11">P3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P1</font> &amp; <font color="Maroon" title="c12">Q3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Q1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c13">P4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">P4</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c14">Q4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Q4</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E1:94"><i><font color="Green" title="E84">A1</font></i></a>, <a class="txt" href="matrix13.html#E7:94"><i><font color="Green" title="E87">A5</font></i></a>, <a class="txt" href="matrix13.html#E13:94"><i><font color="Green" title="E90">A8</font></i></a>, <a class="txt" href="matrix13.html#E15:94"><i><font color="Green" title="E91">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/>
<a NAME="E17:94"/><b>then </b><i><font color="Green" title="E92">A10</font></i>: 
(  <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P1</font>,<font color="Maroon" title="c6">Q1</font></span>)</span>,<font color="Maroon" title="c13">P4</font>,<font color="Maroon" title="c14">Q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c11">P3</font>,<font color="Maroon" title="c12">Q3</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c13">P4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">P2</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c14">Q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">P2</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c13">P4</font>,<font color="Maroon" title="c14">Q4</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="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P1</font>,<font color="Maroon" title="c6">Q1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E3:94"><i><font color="Green" title="E85">A3</font></i></a>, <a class="txt" href="matrix13.html#E9:94"><i><font color="Green" title="E88">A6</font></i></a>, <a class="ref" href="matrix13.html#T56" target="_self" title="MATRIX13:th.56">TT56</a>, <a class="txt" href="matrix13.html#E58"><i><font color="Green" title="E54">LM1</font></i></a></i>;<br/>
 <a href="matrix13.html#K5" title="MATRIX13:func.5">EqSegm</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c9">P2</font>,<font color="Maroon" title="c10">Q2</font> = 
 <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P1</font>,<font color="Maroon" title="c6">Q1</font></span>)</span>,<font color="Maroon" title="c13">P4</font>,<font color="Maroon" title="c14">Q4</font>
<b>by </b><i><a class="txt" href="matrix13.html#E10:94"><i><font color="Green" title="E89">A7</font></i></a>, <a class="txt" href="matrix13.html#E3:94"><i><font color="Green" title="E85">A3</font></i></a>, <a class="txt" href="matrix13.html#E17:94"><i><font color="Green" title="E92">A10</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def4</a></i>
<br/>.= 
 <a href="matrix13.html#K5" title="MATRIX13:func.5">EqSegm</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P1</font>,<font color="Maroon" title="c6">Q1</font></span>)</span>,<font color="Maroon" title="c13">P4</font>,<font color="Maroon" title="c14">Q4</font>
<b>by </b><i><a class="txt" href="matrix13.html#E17:94"><i><font color="Green" title="E92">A10</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def4</a></i>
;<br/>
<b>hence </b><a NAME="E19:94"/>
 <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P1</font>,<font color="Maroon" title="c6">Q1</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E4:94"><i><font color="Green" title="E86">A4</font></i></a>, <a class="txt" href="matrix13.html#E17:94"><i><font color="Green" title="E92">A10</font></i></a>, <a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def5</a></i>;<br/>


</div>
