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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">M</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font>  st  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> holds <br/><font color="Olive" title="b2">M</font> is  <a href="matrix13.html#NV1" title="MATRIX13:NV.1">without_repeated_line</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">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#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">K</font>  st  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> holds <br/><font color="Olive" title="b1">M</font> is  <a href="matrix13.html#NV1" title="MATRIX13:NV.1">without_repeated_line</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">M</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">K</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> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> implies <font color="Maroon" title="c4">M</font> is  <a href="matrix13.html#NV1" title="MATRIX13:NV.1">without_repeated_line</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:129"/><span class="lab"><font color="Green" title="E104">A1</font></span>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 ; <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="c4">M</font> is  <a href="matrix13.html#NV1" title="MATRIX13:NV.1">without_repeated_line</a> </span><br/>

<a NAME="E2:129"/><span class="lab"><font color="Green" title="E105">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:129"/>
not <font color="Maroon" title="c4">M</font> is  <a href="matrix13.html#NV1" title="MATRIX13:NV.1">without_repeated_line</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:129"/><span class="lab"><font color="Green" title="E106">A3</font></span>: 
<font color="Maroon" title="c5">x1</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="c4">M</font>
 <span class="kw">and </span><br/><a NAME="E6:129"/><span class="lab"><font color="Green" title="E107">A4</font></span>: 
<font color="Maroon" title="c6">x2</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="c4">M</font>
 <span class="kw">and </span><br/><a NAME="E7:129"/><span class="lab"><font color="Green" title="E108">A5</font></span>: 
<font color="Maroon" title="c4">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font>
 <span class="kw">and </span><br/><a NAME="E8:129"/><span class="lab"><font color="Green" title="E109">A6</font></span>: 
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c8">x2</font> = <font color="Maroon" title="c6">x2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:129"><span class="lab"><font color="Green" title="E106">A3</font></span></a>, <a class="txt" href="#E6:129"><span class="lab"><font color="Green" title="E107">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E11:129"/><span class="lab"><font color="Green" title="E110">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E12:129"/><span class="lab"><font color="Green" title="E111">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K1" title="MATRLIN:func.1">Del</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c7">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:129"><span class="lab"><font color="Green" title="E106">A3</font></span></a>, <a class="ref" href="finseq_3.html#T113" title="FINSEQ_3:th.113">FINSEQ_3:113</a></span>;<br/>
<a NAME="E13:129"/><span class="lab"><font color="Green" title="E112">A9</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">M</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="c4">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E14:129"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A10</font></span>: 
<font color="Maroon" title="c4">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c8">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:129"><span class="lab"><font color="Green" title="E107">A4</font></span></a>, <a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E105">A2</font></span></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></span>;<br/>
<a NAME="E15:129"/>
<font color="Maroon" title="c4">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c7">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:129"><span class="lab"><font color="Green" title="E106">A3</font></span></a>, <a class="txt" href="#E13:129"><span class="lab"><font color="Green" title="E112">A9</font></span></a>, <a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E105">A2</font></span></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c4">M</font> = 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c7">x1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c8">x2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:129"><span class="lab"><font color="Green" title="E108">A5</font></span></a>, <a class="txt" href="#E14:129"><span class="lab"><font color="Green" title="E113">A10</font></span></a>, <a class="ref" href="matrix11.html#T30" title="MATRIX11:th.30">MATRIX11:30</a></span>
<br/>.= 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c7">x1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c8">x2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T70" title="FVSUM_1:th.70">FVSUM_1:70</a></span>
;<br/>
<a NAME="E17:129"/><span class="kw">then </span>
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c7">x1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129"><span class="lab"><font color="Green" title="E104">A1</font></span></a>, <a class="txt" href="#E6:129"><span class="lab"><font color="Green" title="E107">A4</font></span></a>, <a class="txt" href="#E8:129"><span class="lab"><font color="Green" title="E109">A6</font></span></a>, <a class="txt" href="#E13:129"><span class="lab"><font color="Green" title="E112">A9</font></span></a>, <a class="ref" href="matrix13.html#T93" target="_self" title="MATRIX13:th.93">Th93</a></span>;<br/>
<a NAME="E18:129"/><span class="kw">then </span>
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:129"><span class="lab"><font color="Green" title="E111">A8</font></span></a>, <a class="ref" href="matrix13.html#T74" target="_self" title="MATRIX13:th.74">Th74</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:129"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:129"><span class="lab"><font color="Green" title="E105">A2</font></span></a>, <a class="txt" href="#E11:129"><span class="lab"><font color="Green" title="E110">A7</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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/>


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