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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M</font> being  <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">P</font>, <font color="Olive" title="b3">P1</font>, <font color="Olive" title="b4">Q</font>, <font color="Olive" title="b5">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>   st <font color="Olive" title="b2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">P1</font> &amp; <font color="Olive" title="b4">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">Q1</font> holds <br/> <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">P</font>,<font color="Olive" title="b4">Q</font></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> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Olive" title="b1">M</font>,<font color="Olive" title="b3">P1</font>,<font color="Olive" title="b5">Q1</font></span>)</span></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">P1</font>, <font color="Olive" title="b3">Q</font>, <font color="Olive" title="b4">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>   st <font color="Olive" title="b1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">P1</font> &amp; <font color="Olive" title="b3">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">Q1</font> holds <br/> <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Olive" title="b1">P</font>,<font color="Olive" title="b3">Q</font></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> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Olive" title="b2">P1</font>,<font color="Olive" title="b4">Q1</font></span>)</span></span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> implies  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">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#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">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> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:95"/><span class="lab"><font color="Green" title="E84">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:95"/><span class="lab"><font color="Green" title="E85">A2</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">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#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">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></span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">S1</font> =  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P1</font>,<font color="Maroon" title="c6">Q1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">S</font> =  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c5">Q</font>;<br/>
<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E4:95"/><span class="lab"><font color="Green" title="E86">A3</font></span>: 
<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#K6" title="MATRIX13:func.6">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>
 <span class="kw">and </span><br/><a NAME="E5:95"/><span class="lab"><font color="Green" title="E87">A4</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">P2</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="c10">Q2</font>
 <span class="kw">and </span><br/><a NAME="E6:95"/><span class="lab"><font color="Green" title="E88">A5</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">P2</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="matrix13.html#K6" title="MATRIX13:func.6">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>
 <span class="kw">and </span><br/><a NAME="E7:95"/><span class="lab"><font color="Green" title="E89">A6</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></span>;<br/>
<a NAME="E8:95"/>
( <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:95"><span class="lab"><font color="Green" title="E87">A4</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E10:95"/><span class="lab"><font color="Green" title="E90">A7</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E11:95"/><span class="lab"><font color="Green" title="E91">A8</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E12:95"/>
<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#K5" title="MATRIX13:func.5">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>
 <span class="kw">and </span><br/><a NAME="E13:95"/>
<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#K5" title="MATRIX13:func.5">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>
 <span class="kw">and </span><br/><a NAME="E14:95"/><span class="lab"><font color="Green" title="E92">A9</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">P3</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="c9">P2</font>
 <span class="kw">and </span><br/><a NAME="E15:95"/><span class="lab"><font color="Green" title="E93">A10</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">Q3</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="c10">Q2</font>
 <span class="kw">and </span><br/><a NAME="E16:95"/><span class="lab"><font color="Green" title="E94">A11</font></span>: 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">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#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c11">P3</font>,<font color="Maroon" title="c12">Q3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:95"><span class="lab"><font color="Green" title="E86">A3</font></span></a>, <a class="ref" href="matrix13.html#T57" target="_self" title="MATRIX13:th.57">Th57</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">P4</font> = <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">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#K5" title="MATRIX13:func.5">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T53" target="_self" title="MATRIX13:th.53">Th53</a></span>;<br/>
<a NAME="E18:95"/><span class="lab"><font color="Green" title="E95">A12</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">Q4</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="c9">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:95"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="txt" href="#E5:95"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E11:95"><span class="lab"><font color="Green" title="E91">A8</font></span></a>, <a class="txt" href="#E15:95"><span class="lab"><font color="Green" title="E93">A10</font></span></a>, <a class="txt" href="matrix13.html#E58"><span class="lab"><font color="Green" title="E54">Lm2</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E19:95"/><span class="lab"><font color="Green" title="E96">A13</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">P4</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="c9">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E10:95"><span class="lab"><font color="Green" title="E90">A7</font></span></a>, <a class="txt" href="#E14:95"><span class="lab"><font color="Green" title="E92">A9</font></span></a>, <a class="txt" href="matrix13.html#E58"><span class="lab"><font color="Green" title="E54">Lm2</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E20:95"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="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="Olive" title="b1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">Th43</a></span>;<br/>
<a NAME="E21:95"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A14</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c14">Q4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Q4</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/>
<a NAME="E22:95"/><span class="lab"><font color="Green" title="E98">A15</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:95"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="txt" href="#E11:95"><span class="lab"><font color="Green" title="E91">A8</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E23:95"/><span class="lab"><font color="Green" title="E99">A16</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E10:95"><span class="lab"><font color="Green" title="E90">A7</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E24:95"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="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="Olive" title="b1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">Th43</a></span>;<br/>
<a NAME="E25:95"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c13">P4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">P4</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/>
<a NAME="E26:95"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A17</font></span>: 
<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#K6" title="MATRIX13:func.6">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:95"><span class="lab"><font color="Green" title="E99">A16</font></span></a>, <a class="txt" href="#E22:95"><span class="lab"><font color="Green" title="E98">A15</font></span></a>, <a class="txt" href="#E21:95"><span class="lab"><font color="Green" title="E97">A14</font></span></a>, <a class="ref" href="matrix13.html#T56" target="_self" title="MATRIX13:th.56">Th56</a></span>;<br/>
<a NAME="E27:95"/>
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">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#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c11">P3</font>,<font color="Maroon" title="c12">Q3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:95"><span class="lab"><font color="Green" title="E99">A16</font></span></a>, <a class="txt" href="#E22:95"><span class="lab"><font color="Green" title="E98">A15</font></span></a>, <a class="ref" href="matrix13.html#T56" target="_self" title="MATRIX13:th.56">Th56</a></span>;<br/>
<span class="kw">then </span> <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">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#K6" title="MATRIX13:func.6">Segm</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:95"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E16:95"><span class="lab"><font color="Green" title="E94">A11</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></span>
<br/>.= 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:95"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E15:95"><span class="lab"><font color="Green" title="E93">A10</font></span></a>, <a class="txt" href="#E22:95"><span class="lab"><font color="Green" title="E98">A15</font></span></a>, <a class="txt" href="#E19:95"><span class="lab"><font color="Green" title="E96">A13</font></span></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a>, <a class="txt" href="matrix13.html#E58"><span class="lab"><font color="Green" title="E54">Lm2</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E29:95"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">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#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:95"><span class="lab"><font color="Green" title="E88">A5</font></span></a>, <a class="txt" href="#E7:95"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="txt" href="#E19:95"><span class="lab"><font color="Green" title="E96">A13</font></span></a>, <a class="txt" href="#E18:95"><span class="lab"><font color="Green" title="E95">A12</font></span></a>, <a class="txt" href="#E26:95"><span class="lab"><font color="Green" title="E100">A17</font></span></a>, <a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
