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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  <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  <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"> (  <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">assume </span><a NAME="E1:27"/><span class="lab"><font color="Green" title="E12">A1</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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:27_1"/>
( <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>  or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1"><span class="kw">suppose </span></a><a NAME="E1:27_1_1"/><span class="lab"><font color="Green" title="E13">A2</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> 
 ; <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/><div class="add"><a NAME="E2:27_1_1"/><span class="kw">then </span>
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>
 <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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">ONE</font> =  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">m</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_1"><span class="lab"><font color="Green" title="E13">A2</font></span></a></span>;<br/><a NAME="E4:27_1_1"/>
 <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">m</font>) is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_6.html#T8" title="MATRIX_6:th.8">MATRIX_6:8</a></span>;<br/><a NAME="E5:27_1_1"/><span class="kw">then </span>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">m</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="c3">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="laplace.html#T34" title="LAPLACE:th.34">LAPLACE:34</a></span>;<br/><a NAME="E6:27_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A3</font></span>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c5">ONE</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="matrix13.html#T83" title="MATRIX13:th.83">MATRIX13:83</a></span>;<br/><a NAME="E7:27_1_1"/>
 <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="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_1"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="matrix_1.html#D2" title="MATRIX_1:def.2">MATRIX_1:def 2</a></span>;<br/><a NAME="E8:27_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ;<br/><a NAME="E9:27_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">A</font> <a href="matrlin.html#K8" title="MATRLIN:func.8">^</a> <font color="Maroon" title="c5">ONE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">ONE</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>;<br/><span class="kw">hence </span><a NAME="E10:27_1_1"/>
 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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27_1_1"><span class="lab"><font color="Green" title="E14">A3</font></span></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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2"><span class="kw">suppose </span></a><a NAME="E1:27_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <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/><div class="add"><span class="kw">hence </span><a NAME="E2:27_1_2"/>
 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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="matrtop1.html#E18"><span class="lab"><font color="Green" title="E10">Lm1</font></span></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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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