<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c3">W</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c1">m</font> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c2">K</font>;<br/>





<b>consider </b><font color="Maroon" title="c4">I</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">W</font><b> such that </b><br/><a NAME="E2:87"/><i><font color="Green" title="E69">A1</font></i>: 
<font color="Maroon" title="c4">I</font> is    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#D1" title="VECTSP_9:def.1">VECTSP_9:def 1</a></i>;<br/>
<a NAME="E3:87"/><i><font color="Green" title="E70">A2</font></i>: 
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W</font> &amp; <font color="Maroon" title="c4">I</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">W</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c3">W</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c3">W</font> #) )
 
<b>by </b><i><a class="txt" href="matrix15.html#E2:87"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="vectsp_9.html#D2" title="VECTSP_9:def.2">VECTSP_9:def 2</a>, <a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">U</font> = <font color="Maroon" title="c4">I</font> as  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c2">K</font></span>)</span> <b>by </b><i><a class="ref" href="vectsp_9.html#T15" title="VECTSP_9:th.15">VECTSP_9:15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">M</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E6:87"/><i><font color="Green" title="E71">A3</font></i>: 
<font color="Maroon" title="c6">M</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E7:87"/><i><font color="Green" title="E72">A4</font></i>: 
 <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">U</font>
 <b>by </b><i><a class="ref" href="matrix13.html#T104" title="MATRIX13:th.104">MATRIX13:104</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">K</font>,  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font>) <b>-&gt; </b>  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">K</font> = $1;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">K</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> holds <br/>( <font color="Olive" title="b1">A</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Olive" title="b1">A</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">W</font> );<br/>
<a NAME="E8:87"/><i><font color="Green" title="E73">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c6">M</font>]
 
<b>by </b><i><a class="txt" href="matrix15.html#E3:87"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="vectsp_9.html#T21" title="VECTSP_9:th.21">VECTSP_9:21</a>, <a class="txt" href="matrix15.html#E6:87"><i><font color="Green" title="E71">A3</font></i></a>, <a class="txt" href="matrix15.html#E7:87"><i><font color="Green" title="E72">A4</font></i></a></i>;<br/>
<a NAME="E9:87"/><i><font color="Green" title="E74">A6</font></i>: 
for <font color="Olive" title="b1">A'</font>, <font color="Olive" title="b2">B'</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">K</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">B'</font>] holds <br/>for <font color="Olive" title="b3">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font><br/> for <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">j</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> <font color="Olive" title="b1">A'</font> &amp; ( <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j</font> implies <font color="Olive" title="b3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> ) holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Olive" title="b1">A'</font>,<font color="Olive" title="b4">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="Olive" title="b1">A'</font>,<font color="Olive" title="b4">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Olive" title="b1">A'</font>,<font color="Olive" title="b5">j</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">B'</font>,<font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font>,<font color="Olive" title="b3">a</font>)]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">A'</font>, <font color="Maroon" title="c8">B'</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">K</font>;<br/>



<b>assume </b><a NAME="E1:87_1"/><i><font color="Green" title="E75">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c8">B'</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c9">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font>;<br/><b>let </b><font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c11">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>





<b>assume </b><a NAME="E2:87_1"/><i><font color="Green" title="E76">A8</font></i>: 
( <font color="Maroon" title="c11">j</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> <font color="Maroon" title="c7">A'</font> &amp; ( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j</font> implies <font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> ) )
 ;<br/>

<b>set </b><font color="Maroon" title="c12">R</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c10">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">A'</font>,<font color="Maroon" title="c10">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="c9">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">A'</font>,<font color="Maroon" title="c11">j</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E3:87_1"/><i><font color="Green" title="E77">A9</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">A'</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">A'</font></span>)</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/>
<a NAME="E4:87_1"/><i><font color="Green" title="E78">A10</font></i>: 
( <font color="Maroon" title="c7">A'</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c7">A'</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> )
 
<b>by </b><i><a class="txt" href="matrix15.html#E1:87_1"><i><font color="Green" title="E75">A7</font></i></a></i>;<br/>
<a NAME="E5:87_1"/><b>then </b><i><font color="Green" title="E79">A11</font></i>: 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c7">A'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c10">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">A'</font>,<font color="Maroon" title="c10">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="c9">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">A'</font>,<font color="Maroon" title="c11">j</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix15.html#T69" target="_self" title="MATRIX15:th.69">Th69</a>, <a class="txt" href="matrix15.html#E2:87_1"><i><font color="Green" title="E76">A8</font></i></a></i>;<br/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font> = 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">A'</font>
<b>by </b><i><a class="txt" href="matrix15.html#E4:87_1"><i><font color="Green" title="E78">A10</font></i></a>, <a class="ref" href="matrix13.html#T121" title="MATRIX13:th.121">MATRIX13:121</a></i>
<br/>.= 
 <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">A'</font>,<font color="Maroon" title="c10">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">A'</font>,<font color="Maroon" title="c10">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="c9">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">A'</font>,<font color="Maroon" title="c11">j</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix15.html#E2:87_1"><i><font color="Green" title="E76">A8</font></i></a>, <a class="txt" href="matrix15.html#E3:87_1"><i><font color="Green" title="E77">A9</font></i></a>, <a class="ref" href="matrix13.html#T92" title="MATRIX13:th.92">MATRIX13:92</a></i>
;<br/>
<b>hence </b><a NAME="E7:87_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c10">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">A'</font>,<font color="Maroon" title="c10">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="c9">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">A'</font>,<font color="Maroon" title="c11">j</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">B'</font>,<font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c9">a</font>)]
 <b>by </b><i><a class="txt" href="matrix15.html#E1:87_1"><i><font color="Green" title="E75">A7</font></i></a>, <a class="txt" href="matrix15.html#E5:87_1"><i><font color="Green" title="E79">A11</font></i></a>, <a class="ref" href="matrix13.html#T121" title="MATRIX13:th.121">MATRIX13:121</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c7">A'</font>, <font color="Maroon" title="c8">B'</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">K</font>, <font color="Maroon" title="c9">N</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="E11:87"/><i><font color="Green" title="E75">A12</font></i>: 
<font color="Maroon" title="c9">N</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="c1">m</font>
 <b>and </b><br/><a NAME="E12:87"/><i><font color="Green" title="E76">A13</font></i>: 
(  <a href="matrix13.html#K8" title="MATRIX13:func.8">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#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c7">A'</font> &amp;  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c6">M</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">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c8">B'</font>] )
 <b>and </b><br/><a NAME="E13:87"/><i><font color="Green" title="E77">A14</font></i>: 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c7">A'</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">N</font></span>)</span></span>)</span>,<font color="Maroon" title="c9">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">N</font></span>)</span>
 <b>and </b><br/><a NAME="E14:87"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</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> <font color="Maroon" title="c7">A'</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">N</font> holds <br/> <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">A'</font>,<font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <b>and </b><br/><a NAME="E15:87"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">N</font></span>)</span> &amp; <font color="Olive" title="b2">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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">A'</font></span>)</span> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c9">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c7">A'</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</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="c2">K</font>
 <b>from </b><i><a class="ref" href="matrix15.html#S2" target="_self" title="MATRIX15:sch.2">MATRIX15:sch 2</a>(<a class="txt" href="matrix15.html#E8:87"><i><font color="Green" title="E73">A5</font></i></a>, <a class="txt" href="matrix15.html#E9:87"><i><font color="Green" title="E74">A6</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c10">A'</font> = <font color="Maroon" title="c7">A'</font> as   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">K</font> <b>by </b><i><a class="txt" href="matrix15.html#E2:87"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="vectsp_9.html#D2" title="VECTSP_9:def.2">VECTSP_9:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">A'</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c9">N</font>
;<br/>

<a NAME="E17:87"/><i><font color="Green" title="E78">A15</font></i>: 
(  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c6">M</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="c4">I</font> &amp;  <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c10">A'</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c10">A'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">W</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c3">W</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c3">W</font> #) )
 
<b>by </b><i><a class="txt" href="matrix15.html#E12:87"><i><font color="Green" title="E76">A13</font></i></a>, <a class="txt" href="matrix15.html#E6:87"><i><font color="Green" title="E71">A3</font></i></a>, <a class="txt" href="matrix15.html#E7:87"><i><font color="Green" title="E72">A4</font></i></a>, <a class="ref" href="matrix13.html#T121" title="MATRIX13:th.121">MATRIX13:121</a></i>;<br/>
<a NAME="E18:87"/>
 <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c10">A'</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 <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c10">A'</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:87_2"/><i><font color="Green" title="E79">A16</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c10">A'</font>
 ;<br/>

<a NAME="E2:87_2"/>
<font color="Maroon" title="c11">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c10">A'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix15.html#E1:87_2"><i><font color="Green" title="E79">A16</font></i></a>, <a class="ref" href="vectsp_7.html#T13" title="VECTSP_7:th.13">VECTSP_7:13</a></i>;<br/>
<b>hence </b><a NAME="E3:87_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c10">A'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c11">lA</font> =  <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c10">A'</font> as  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">W</font> <b>by </b><i><a class="txt" href="matrix15.html#E17:87"><i><font color="Green" title="E78">A15</font></i></a>, <a class="ref" href="vectsp_9.html#T16" title="VECTSP_9:th.16">VECTSP_9:16</a></i>;<br/>
<a NAME="E20:87"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c11">lA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c10">A'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T21" title="VECTSP_9:th.21">VECTSP_9:21</a></i>;<br/>
<b>hence </b><a NAME="E21:87"/>
( <font color="Maroon" title="c9">N</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="c1">m</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W</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">N</font> &amp;  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c10">A'</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W</font></span>)</span></span>)</span>,<font color="Maroon" title="c9">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W</font></span>)</span> &amp;  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c10">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W</font> &amp;  <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c10">A'</font> is    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c3">W</font> )
 <b>by </b><i><a class="txt" href="matrix15.html#E3:87"><i><font color="Green" title="E70">A2</font></i></a>, <a class="txt" href="matrix15.html#E11:87"><i><font color="Green" title="E75">A12</font></i></a>, <a class="txt" href="matrix15.html#E12:87"><i><font color="Green" title="E76">A13</font></i></a>, <a class="txt" href="matrix15.html#E13:87"><i><font color="Green" title="E77">A14</font></i></a>, <a class="txt" href="matrix15.html#E17:87"><i><font color="Green" title="E78">A15</font></i></a>, <a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>


</div>
