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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="firebrick">::  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">A</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Olive" title="b1">K</font>  st ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp;  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> holds <br/> ex <font color="Olive" title="b3">B</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Olive" title="b1">K</font> st  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="matrlin.html#K8" title="MATRLIN:func.8">^</a> <font color="Olive" title="b3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font></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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">K</font>  st ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp;  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> holds <br/> ex <font color="Olive" title="b2">B</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">K</font> st  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrlin.html#K8" title="MATRLIN:func.8">^</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp;  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> implies  ex <font color="Olive" title="b1">B</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">K</font> st  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="matrlin.html#K8" title="MATRLIN:func.8">^</a> <font color="Olive" title="b1">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">V</font> = <font color="Maroon" title="c2">m</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c3">K</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">L</font> =  <a href="matrix13.html#K9" title="MATRIX13:func.9">lines</a> <font color="Maroon" title="c4">A</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">and </span><br/><a NAME="E2:26"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">K</font> st  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="matrlin.html#K8" title="MATRLIN:func.8">^</a> <font color="Olive" title="b1">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font></span><br/>

<a NAME="E3:26"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="matrix13.html#T1" title="MATRIX13:th.1">MATRIX13:1</a></span>;<br/>
<a NAME="E4:26"/>
 <a href="matrix13.html#K9" title="MATRIX13:func.9">lines</a> <font color="Maroon" title="c4">A</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="matrix13.html#T121" title="MATRIX13:th.121">MATRIX13:121</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c3">K</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:26"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
 <a href="matrix13.html#K9" title="MATRIX13:func.9">lines</a> <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">and </span><br/><a NAME="E7:26"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<font color="Maroon" title="c7">B</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> 
 <span class="kw">and </span><br/><a NAME="E8:26"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c7">B</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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c3">K</font></span>)</span>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c3">K</font></span>)</span>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c3">K</font></span>)</span>, the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c3">K</font></span>)</span> #)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#T17" title="VECTSP_7:th.17">VECTSP_7:17</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">B</font> = <font color="Maroon" title="c7">B</font> as   <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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c3">K</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="vectsp_9.html#T21" title="VECTSP_9:th.21">VECTSP_9:21</a></span>;<br/>
<a NAME="E10:26"/>
<font color="Maroon" title="c8">B</font> is    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c2">m</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E17">A7</font></span>:  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">B</font> = 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c3">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#D1" title="VECTSP_9:def.1">VECTSP_9:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c2">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T112" title="MATRIX13:th.112">MATRIX13:112</a></span>
;<br/>
<a NAME="E12:26"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">A</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="txt" href="#E1:26"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="matrix13.html#T1" title="MATRIX13:th.1">MATRIX13:1</a></span>;<br/>
<a NAME="E13:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A8</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="matrix13.html#T74" title="MATRIX13:th.74">MATRIX13:74</a>, <a class="ref" href="xreal_1.html#T48" title="XREAL_1:th.48">XREAL_1:48</a></span>;<br/>
<a NAME="E14:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A9</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<a NAME="E15:26"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
<font color="Maroon" title="c4">A</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="matrix13.html#T121" title="MATRIX13:th.121">MATRIX13:121</a></span>;<br/>
<a NAME="E16:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">lines</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T62" title="FINSEQ_4:th.62">FINSEQ_4:62</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">BL</font> = <font color="Maroon" title="c8">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">lines</a> <font color="Maroon" title="c4">A</font></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">p</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E18:26"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">lines</a> <font color="Maroon" title="c4">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E19:26"/><span class="lab"><font color="Green" title="E23">A13</font></span>: 
<font color="Maroon" title="c10">p</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T58" title="FINSEQ_4:th.58">FINSEQ_4:58</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">p</font> = <font color="Maroon" title="c10">p</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c3">K</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:26"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p</font> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">lines</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:26"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E19:26"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="ref" href="finseq_4.html#T62" title="FINSEQ_4:th.62">FINSEQ_4:62</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">lines</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a></span>
<br/>.= 
<font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E16:26"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E11:26"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">P</font> =  <a href="matrix13.html#K10" title="MATRIX13:func.10">FinS2MX</a> <font color="Maroon" title="c11">p</font> as   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">K</font> ;<br/>
<a NAME="E23:26"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:26"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/>
<a NAME="E24:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A14</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="matrlin.html#K8" title="MATRLIN:func.8">^</a> <font color="Maroon" title="c12">P</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E19:26"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="ref" href="finseq_3.html#T91" title="FINSEQ_3:th.91">FINSEQ_3:91</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">P</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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"><font color="Maroon" title="c4">A</font> <a href="matrlin.html#K8" title="MATRLIN:func.8">^</a> <font color="Maroon" title="c12">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font></span><br/>

 <a href="matrix13.html#K9" title="MATRIX13:func.9">lines</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="matrlin.html#K8" title="MATRLIN:func.8">^</a> <font color="Maroon" title="c12">P</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">lines</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T31" title="FINSEQ_1:th.31">FINSEQ_1:31</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">lines</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:26"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>
<br/>.= 
<font color="Maroon" title="c8">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E26:26"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="matrlin.html#K8" title="MATRLIN:func.8">^</a> <font color="Maroon" title="c12">P</font></span>)</span> <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="txt" href="#E7:26"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E24:26"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="ref" href="matrix13.html#T121" title="MATRIX13:th.121">MATRIX13:121</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
