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

<span class="kw">let </span><font color="Maroon" title="c1">n</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">L</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">K</font> holds <br/> (  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b2">L</font>,<font color="Maroon" title="c1">n</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  iff ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Olive" title="b2">L</font> <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="Olive" title="b1">K</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">L</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font> holds <br/> (  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b1">L</font>,<font color="Maroon" title="c1">n</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  iff ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Olive" title="b1">L</font> <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="c2">K</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">L</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">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="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c1">n</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  iff ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c3">L</font> <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="c2">K</font> ) )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">B</font> =  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E1:6"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
(  not  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c1">n</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  or <font color="Maroon" title="c3">L</font> <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="c2">K</font> or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:6_1"/>
 <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c1">n</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</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">L</font> <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="c2">K</font> or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )</span><br/>

<a NAME="E2:6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A2</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c1">n</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="c2">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/>
<span class="kw">assume </span><a NAME="E3:6_1"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c3">L</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>
 ; <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="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><br/>

<span class="kw">assume </span><a NAME="E4:6_1"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</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/>

<a NAME="E5:6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c1">n</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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/>
<a NAME="E6:6_1"/><span class="kw">then </span>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T71" title="FINSEQ_2:th.71">FINSEQ_2:71</a>, <a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a></span>;<br/>
<span class="kw">then </span> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> = 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <font color="Maroon" title="c3">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E5:6_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="fvsum_1.html#T107" title="FVSUM_1:th.107">FVSUM_1:107</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixj1.html#T5" title="MATRIXJ1:th.5">MATRIXJ1:5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:6_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6_1"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="matrixj2.html#T2" target="_self" title="MATRIXJ2:th.2">Th2</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><span class="kw">end;</span></div>
<a NAME="E2:6"/>
( ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c3">L</font> <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="c2">K</font> ) implies  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c1">n</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:6_2"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c3">L</font> <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="c2">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="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c1">n</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> </span><br/>

<span class="kw">assume </span><a NAME="E2:6_2"/>
not  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c1">n</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</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> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> = 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c1">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="laplace.html#T34" title="LAPLACE:th.34">LAPLACE:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixj2.html#T2" target="_self" title="MATRIXJ2:th.2">Th2</a></span>
<br/>.= 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <font color="Maroon" title="c3">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixj1.html#T5" title="MATRIXJ1:th.5">MATRIXJ1:5</a></span>
;<br/>
<a NAME="E4:6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A6</font></span>: 
 ex <font color="Olive" title="b1">k</font> being   <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>  st <br/>( <font color="Olive" title="b1">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <font color="Maroon" title="c3">L</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T107" title="FVSUM_1:th.107">FVSUM_1:107</a></span>;<br/>
<a NAME="E5:6_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:6_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_2"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E4:6_2"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="finseq_2.html#T71" title="FINSEQ_2:th.71">FINSEQ_2:71</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:6"/>
(  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c1">n</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  iff ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c3">L</font> <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="c2">K</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E6">A1</font></span></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>
