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<b>let </b><font color="Maroon" title="c1">K</font> be  <a href="polynom5.html#V2" title="POLYNOM5:attr.2">algebraic-closed</a> <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b1">V</font>,<font color="Olive" title="b1">V</font>  ex <font color="Olive" title="b3">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font> ex <font color="Olive" title="b4">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b1">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b2">F</font>,<font color="Olive" title="b4">b1</font>,<font color="Olive" title="b4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b3">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b5">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b5">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b2">F</font> iff  ex <font color="Olive" title="b6">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b6">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="Olive" title="b3">J</font> &amp; <font color="Olive" title="b3">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b5">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b6">i</font></span>)</span></span>)</span> ) ) ) )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">V</font> being non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font>  st  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b1">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/>for <font color="Olive" title="b2">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b1">V</font>,<font color="Olive" title="b1">V</font>  ex <font color="Olive" title="b3">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font> ex <font color="Olive" title="b4">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b1">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b2">F</font>,<font color="Olive" title="b4">b1</font>,<font color="Olive" title="b4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b3">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b5">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b5">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b2">F</font> iff  ex <font color="Olive" title="b6">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b6">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="Olive" title="b3">J</font> &amp; <font color="Olive" title="b3">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b5">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b6">i</font></span>)</span></span>)</span> ) ) ) );<br/>
<a NAME="E1:53"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">V</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies for <font color="Olive" title="b1">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c2">V</font>  ex <font color="Olive" title="b2">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font> ex <font color="Olive" title="b3">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c2">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">b1</font>,<font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b2">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b4">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b4">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b1">F</font> iff  ex <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b5">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="Olive" title="b2">J</font> &amp; <font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b4">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font></span>)</span></span>)</span> ) ) ) ) )</span><br/>

<b>assume </b><a NAME="E1:53_1"/><i><font color="Green" title="E39">A2</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c2">V</font>  ex <font color="Olive" title="b2">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font> ex <font color="Olive" title="b3">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c2">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">b1</font>,<font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b2">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b4">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b4">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b1">F</font> iff  ex <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b5">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="Olive" title="b2">J</font> &amp; <font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b4">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font></span>)</span></span>)</span> ) ) ) )</span><br/>

<a NAME="E2:53_1"/>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:53_1"><i><font color="Green" title="E39">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:53_1"/>
for <font color="Olive" title="b1">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c2">V</font>  ex <font color="Olive" title="b2">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font> ex <font color="Olive" title="b3">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c2">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">b1</font>,<font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b2">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b4">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b4">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b1">F</font> iff  ex <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b5">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="Olive" title="b2">J</font> &amp; <font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b4">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font></span>)</span></span>)</span> ) ) ) )
 <b>by </b><i><a class="ref" href="matrlin2.html#T42" title="MATRLIN2:th.42">MATRLIN2:42</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:53"/><i><font color="Green" title="E39">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:53_2"/><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<b>set </b><font color="Maroon" title="c3">n1</font> = <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<b>let </b><font color="Maroon" title="c4">V</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c4">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies for <font color="Olive" title="b1">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c4">V</font>  ex <font color="Olive" title="b2">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font> ex <font color="Olive" title="b3">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">b1</font>,<font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b2">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b4">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b4">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b1">F</font> iff  ex <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b5">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="Olive" title="b2">J</font> &amp; <font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b4">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font></span>)</span></span>)</span> ) ) ) ) )</span><br/>

<b>assume </b><a NAME="E2:53_2"/><i><font color="Green" title="E41">A5</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c4">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c4">V</font>  ex <font color="Olive" title="b2">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font> ex <font color="Olive" title="b3">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">b1</font>,<font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b2">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b4">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b4">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b1">F</font> iff  ex <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b5">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="Olive" title="b2">J</font> &amp; <font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b4">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font></span>)</span></span>)</span> ) ) ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_2_1"/>
(  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c4">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> or  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="#E2:53_2"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1"><b>suppose </b></a><a NAME="E1:53_2_1_1"/>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c4">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c4">V</font>  ex <font color="Olive" title="b2">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font> ex <font color="Olive" title="b3">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">b1</font>,<font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b2">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b4">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b4">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b1">F</font> iff  ex <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b5">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="Olive" title="b2">J</font> &amp; <font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b4">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font></span>)</span></span>)</span> ) ) ) )</span><br/><div class="add"><b>hence </b><a NAME="E2:53_2_1_1"/>
for <font color="Olive" title="b1">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c4">V</font>  ex <font color="Olive" title="b2">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font> ex <font color="Olive" title="b3">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">b1</font>,<font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b2">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b4">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b4">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b1">F</font> iff  ex <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b5">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="Olive" title="b2">J</font> &amp; <font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b4">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font></span>)</span></span>)</span> ) ) ) )
 <b>by </b><i><a class="txt" href="#E1:53_2"><i><font color="Green" title="E40">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_2"><b>suppose </b></a><a NAME="E1:53_2_1_2"/><i><font color="Green" title="E42">A6</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c4">V</font>  ex <font color="Olive" title="b2">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font> ex <font color="Olive" title="b3">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">b1</font>,<font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b2">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b4">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b4">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b1">F</font> iff  ex <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b5">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="Olive" title="b2">J</font> &amp; <font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b4">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font></span>)</span></span>)</span> ) ) ) )</span><br/><div class="add"><b>let </b><font color="Maroon" title="c5">F</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c4">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font> ex <font color="Olive" title="b2">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c5">F</font>,<font color="Olive" title="b2">b1</font>,<font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b1">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b3">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b3">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font> iff  ex <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b4">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="Olive" title="b1">J</font> &amp; <font color="Olive" title="b1">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b3">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> ) ) ) )</span><br/><a NAME="E2:53_2_1_2"/><i><font color="Green" title="E43">A7</font></i>: 
<font color="Maroon" title="c5">F</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a>
 <b>by </b><i><a class="ref" href="vectsp11.html#T16" title="VECTSP11:th.16">VECTSP11:16</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">V</font>, <font color="Maroon" title="c7">L</font> being   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font><b> such that </b><br/><a NAME="E4:53_2_1_2"/><i><font color="Green" title="E44">A8</font></i>: 
( <font color="Maroon" title="c6">v</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="c4">V</font> &amp; <font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c6">v</font> )
 <b>by </b><i><a class="ref" href="vectsp11.html#D1" title="VECTSP11:def.1">VECTSP11:def 1</a></i>;<br/><b>set </b><font color="Maroon" title="c8">FL</font> = <font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c9">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:53_2_1_2"/><i><font color="Green" title="E45">A9</font></i>: 
 <a href="vectsp11.html#K3" title="VECTSP11:func.3">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp11.html#T27" title="VECTSP11:th.27">VECTSP11:27</a></i>;<br/><b>set </b><font color="Maroon" title="c10">KER</font> =  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c11">IM</font> =  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span>;<br/><a NAME="E7:53_2_1_2"/><i><font color="Green" title="E46">A10</font></i>: 
( <font color="Maroon" title="c7">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> 1 <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:53_2_1_2"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="#E4:53_2_1_2"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="vectsp11.html#T19" title="VECTSP11:th.19">VECTSP11:19</a>, <a class="ref" href="vectsp11.html#D2" title="VECTSP11:def.2">VECTSP11:def 2</a></i>;<br/><a NAME="E8:53_2_1_2"/><b>then </b>
( not  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> &amp;  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:53_2_1_2"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="#E6:53_2_1_2"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="vectsp11.html#T14" title="VECTSP11:th.14">VECTSP11:14</a>, <a class="ref" href="vectsp11.html#T25" title="VECTSP11:th.25">VECTSP11:25</a></i>;<br/><a NAME="E9:53_2_1_2"/><b>then </b><i><font color="Green" title="E47">A11</font></i>: 
(  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span></span>)</span> <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>  &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="matrlin2.html#T42" title="MATRLIN2:th.42">MATRLIN2:42</a>, <a class="ref" href="vectsp_9.html#T29" title="VECTSP_9:th.29">VECTSP_9:29</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">FK</font> = <font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> as   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="vectsp11.html#T29" title="VECTSP11:th.29">VECTSP11:29</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">FI</font> = <font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> as   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="vectsp11.html#T33" title="VECTSP11:th.33">VECTSP11:33</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">Jk</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c7">L</font>,<font color="Maroon" title="c1">K</font>, <font color="Maroon" title="c15">Bker</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span><b> such that </b><br/><a NAME="E13:53_2_1_2"/><i><font color="Green" title="E48">A12</font></i>: 
 <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c12">FK</font>,<font color="Maroon" title="c15">Bker</font>,<font color="Maroon" title="c15">Bker</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c14">Jk</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrixj2.html#T30" target="_self" title="MATRIXJ2:th.30">Th30</a></i>;<br/><a NAME="E14:53_2_1_2"/><i><font color="Green" title="E49">A13</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Jk</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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_2_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Jk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then </b> <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c14">Jk</font> = 
 <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
<b>by </b><i><a class="ref" href="finseq_2.html#T113" title="FINSEQ_2:th.113">FINSEQ_2:113</a></i>
<br/>.= 
 <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 

;<br/>
<b>then </b> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c14">Jk</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixj1.html#D5" title="MATRIXJ1:def.5">MATRIXJ1:def 5</a>, <a class="ref" href="rvsum_1.html#T102" title="RVSUM_1:th.102">RVSUM_1:102</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">Bker</font>
<b>by </b><i><a class="txt" href="#E13:53_2_1_2"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>
<br/>.= 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrlin2.html#T21" title="MATRLIN2:th.21">MATRLIN2:21</a></i>
;<br/>
<b>hence </b><a NAME="E4:53_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E9:53_2_1_2"><i><font color="Green" title="E47">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E15:53_2_1_2"/><i><font color="Green" title="E50">A14</font></i>: 
( <font color="Maroon" title="c4">V</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a>  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span>, <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c4">V</font> )
 <b>by </b><i><a class="txt" href="#E6:53_2_1_2"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="vectsp11.html#T34" title="VECTSP11:th.34">VECTSP11:34</a>, <a class="ref" href="vectsp11.html#T35" title="VECTSP11:th.35">VECTSP11:35</a></i>;<br/><a NAME="E16:53_2_1_2"/><b>then </b><i><font color="Green" title="E51">A15</font></i>: 
(  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p0">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p0">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></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="c4">V</font> &amp;  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span> is    <a href="vectsp_5.html#M1" title="VECTSP_5:mode.1">Linear_Compl</a> of  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span> )
 <b>by </b><i><a class="ref" href="vectsp_5.html#T47" title="VECTSP_5:th.47">VECTSP_5:47</a>, <a class="ref" href="vectsp_5.html#D4" title="VECTSP_5:def.4">VECTSP_5:def 4</a>, <a class="ref" href="vectsp_9.html#T38" title="VECTSP_9:th.38">VECTSP_9:38</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_2_1_2_2"/>
(  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> or  not  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_2_2_1"><b>suppose </b></a><a NAME="E1:53_2_1_2_2_1"/><i><font color="Green" title="E52">A16</font></i>: 
 <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font> ex <font color="Olive" title="b2">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c5">F</font>,<font color="Olive" title="b2">b1</font>,<font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b1">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b3">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b3">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font> iff  ex <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b4">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="Olive" title="b1">J</font> &amp; <font color="Olive" title="b1">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b3">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> ) ) ) )</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c16">Bim</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span>;<br/><a NAME="E3:53_2_1_2_2_1"/>
<font color="Maroon" title="c15">Bker</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">Bim</font> is    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E15:53_2_1_2"><i><font color="Green" title="E50">A14</font></i></a>, <a class="ref" href="matrlin2.html#T26" title="MATRLIN2:th.26">MATRLIN2:26</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">BB</font> = <font color="Maroon" title="c15">Bker</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">Bim</font> as    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">V</font> <b>by </b><i><a class="txt" href="#E16:53_2_1_2"><i><font color="Green" title="E51">A15</font></i></a>, <a class="ref" href="matrlin2.html#T4" title="MATRLIN2:th.4">MATRLIN2:4</a></i>;<br/><b>take </b>
<font color="Maroon" title="c14">Jk</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c5">F</font>,<font color="Olive" title="b1">b1</font>,<font color="Olive" title="b1">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c14">Jk</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b2">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b2">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font> iff  ex <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b3">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="c14">Jk</font> &amp; <font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b2">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b3">i</font></span>)</span></span>)</span> ) ) ) )</span><br/><b>take </b>
<font color="Maroon" title="c17">BB</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c17">BB</font>,<font color="Maroon" title="c17">BB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c14">Jk</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b1">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b1">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font> iff  ex <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b2">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="c14">Jk</font> &amp; <font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b1">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b2">i</font></span>)</span></span>)</span> ) ) ) )</span><br/> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  = 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:53_2_1_2_2_1"><i><font color="Green" title="E52">A16</font></i></a>, <a class="ref" href="matrlin2.html#T42" title="MATRLIN2:th.42">MATRLIN2:42</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">Bim</font>
<b>by </b><i><a class="ref" href="matrlin2.html#T21" title="MATRLIN2:th.21">MATRLIN2:21</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c13">FI</font>,<font color="Maroon" title="c16">Bim</font>,<font color="Maroon" title="c16">Bim</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>
;<br/><a NAME="E6:53_2_1_2_2_1"/><b>then </b><i><font color="Green" title="E53">A17</font></i>: 
 <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c13">FI</font>,<font color="Maroon" title="c16">Bim</font>,<font color="Maroon" title="c16">Bim</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E7:53_2_1_2_2_1"/>
( (  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) &amp; (  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) )
 ;<br/><b>hence </b> <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c17">BB</font>,<font color="Maroon" title="c17">BB</font> = 
 <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <span class="p1"><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c12">FK</font>,<font color="Maroon" title="c15">Bker</font>,<font color="Maroon" title="c15">Bker</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c13">FI</font>,<font color="Maroon" title="c16">Bim</font>,<font color="Maroon" title="c16">Bim</font></span>)</span></span><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
<b>by </b><i><a class="txt" href="#E15:53_2_1_2"><i><font color="Green" title="E50">A14</font></i></a>, <a class="ref" href="matrlin2.html#T27" title="MATRLIN2:th.27">MATRLIN2:27</a></i>
<br/>.= 
 <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <span class="p1"><a href="matrixj1.html#K16" title="MATRIXJ1:func.16">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c12">FK</font>,<font color="Maroon" title="c15">Bker</font>,<font color="Maroon" title="c15">Bker</font></span>)</span></span><a href="matrixj1.html#K16" title="MATRIXJ1:func.16">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:53_2_1_2_2_1"><i><font color="Green" title="E53">A17</font></i></a>, <a class="ref" href="matrixj1.html#T40" title="MATRIXJ1:th.40">MATRIXJ1:40</a></i>
<br/>.= 
 <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c14">Jk</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
<b>by </b><i><a class="txt" href="#E13:53_2_1_2"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="matrixj1.html#T34" title="MATRIXJ1:th.34">MATRIXJ1:34</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b1">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font> iff  ex <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b2">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="c14">Jk</font> &amp; <font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b1">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b2">i</font></span>)</span></span>)</span> ) )</span><br/><b>let </b><font color="Maroon" title="c18">L1</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font> iff  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <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="c14">Jk</font> &amp; <font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c18">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> ) )</span><br/><b>thus </b><a NAME="E9:53_2_1_2_2_1"/>
( <font color="Maroon" title="c18">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font> implies  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <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="c14">Jk</font> &amp; <font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c18">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <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="c14">Jk</font> &amp; <font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c18">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> ) implies <font color="Maroon" title="c18">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_1_2_2_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_2_1_2_2_1_3"/><i><font color="Green" title="E54">A18</font></i>: 
<font color="Maroon" title="c18">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <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="c14">Jk</font> &amp; <font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c18">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> )</span><br/>

<b>take </b><font color="Maroon" title="c19">i</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Jk</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">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="c14">Jk</font> &amp; <font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c18">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span> )</span><br/>

<a NAME="E2:53_2_1_2_2_1_3"/>
<font color="Maroon" title="c19">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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Jk</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E14:53_2_1_2"><i><font color="Green" title="E49">A13</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<b>hence </b><a NAME="E3:53_2_1_2_2_1_3"/><i><font color="Green" title="E55">A19</font></i>: 
<font color="Maroon" title="c19">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="c14">Jk</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c18">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span></span><br/>

<b>consider </b><font color="Maroon" title="c20">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:53_2_1_2_2_1_3"/><i><font color="Green" title="E56">A20</font></i>: 
<font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c7">L</font>,<font color="Maroon" title="c20">k</font>
 <b>by </b><i><a class="txt" href="#E3:53_2_1_2_2_1_3"><i><font color="Green" title="E55">A19</font></i></a>, <a class="ref" href="matrixj2.html#D3" target="_self" title="MATRIXJ2:def.3">Def3</a></i>;<br/>
<a NAME="E6:53_2_1_2_2_1_3"/>
<font color="Maroon" title="c18">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_1_2_2_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_2_1_2_2_1_3_1"/>
<font color="Maroon" title="c7">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">L1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:53_2_1_2_2_1_3_1"/><b>then </b>
( <font color="Maroon" title="c13">FI</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp; <font color="Maroon" title="c18">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c13">FI</font> )
 
<b>by </b><i><a class="txt" href="#E2:53_2_1_2"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="#E7:53_2_1_2"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="#E6:53_2_1_2"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="#E16:53_2_1_2"><i><font color="Green" title="E51">A15</font></i></a>, <a class="txt" href="#E1:53_2_1_2_2_1_3"><i><font color="Green" title="E54">A18</font></i></a>, <a class="ref" href="vectsp11.html#T39" title="VECTSP11:th.39">VECTSP11:39</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">v1</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:53_2_1_2_2_1_3_1"/><i><font color="Green" title="E57">A21</font></i>: 
( <font color="Maroon" title="c21">v1</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> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">FI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">L1</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c21">v1</font> )
 <b>by </b><i><a class="ref" href="vectsp11.html#D2" title="VECTSP11:def.2">VECTSP11:def 2</a></i>;<br/>
<b>thus </b><a NAME="E5:53_2_1_2_2_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:53_2_1_2_2_1_3_1"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="#E1:53_2_1_2_2_1"><i><font color="Green" title="E52">A16</font></i></a>, <a class="ref" href="anproj_1.html#D8" title="ANPROJ_1:def.8">ANPROJ_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:53_2_1_2_2_1_3"/>
<font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c18">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:53_2_1_2_2_1_3"><i><font color="Green" title="E56">A20</font></i></a>, <a class="ref" href="matrixj2.html#D1" target="_self" title="MATRIXJ2:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>given </b><font color="Maroon" title="c19">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E11:53_2_1_2_2_1"/><i><font color="Green" title="E54">A22</font></i>: 
( <font color="Maroon" title="c19">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="c14">Jk</font> &amp; <font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c18">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font></span><br/><b>consider </b><font color="Maroon" title="c20">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E13:53_2_1_2_2_1"/><i><font color="Green" title="E55">A23</font></i>: 
<font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c7">L</font>,<font color="Maroon" title="c20">k</font>
 <b>by </b><i><a class="txt" href="#E11:53_2_1_2_2_1"><i><font color="Green" title="E54">A22</font></i></a>, <a class="ref" href="matrixj2.html#D3" target="_self" title="MATRIXJ2:def.3">Def3</a></i>;<br/><a NAME="E14:53_2_1_2_2_1"/>
<font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E11:53_2_1_2_2_1"><i><font color="Green" title="E54">A22</font></i></a>, <a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>;<br/><a NAME="E15:53_2_1_2_2_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font></span>)</span> <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/><a NAME="E16:53_2_1_2_2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E17:53_2_1_2_2_1"/><b>then </b><i><font color="Green" title="E56">A24</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/><b>then </b><font color="Maroon" title="c7">L</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E13:53_2_1_2_2_1"><i><font color="Green" title="E55">A23</font></i></a>, <a class="ref" href="matrixj2.html#D1" target="_self" title="MATRIXJ2:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c18">L1</font>
<b>by </b><i><a class="txt" href="#E11:53_2_1_2_2_1"><i><font color="Green" title="E54">A22</font></i></a>, <a class="txt" href="#E17:53_2_1_2_2_1"><i><font color="Green" title="E56">A24</font></i></a>, <a class="ref" href="matrixj2.html#D1" target="_self" title="MATRIXJ2:def.1">Def1</a></i>
;<br/><b>hence </b><a NAME="E19:53_2_1_2_2_1"/>
<font color="Maroon" title="c18">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="#E2:53_2_1_2"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="#E4:53_2_1_2"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="vectsp11.html#D2" title="VECTSP11:def.2">VECTSP11:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_2_2_2"><b>suppose </b></a><a NAME="E1:53_2_1_2_2_2"/><i><font color="Green" title="E52">A25</font></i>: 
not  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font> ex <font color="Olive" title="b2">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c5">F</font>,<font color="Olive" title="b2">b1</font>,<font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b1">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b3">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b3">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font> iff  ex <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b4">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="Olive" title="b1">J</font> &amp; <font color="Olive" title="b1">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b3">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> ) ) ) )</span><br/><div class="add"><a NAME="E2:53_2_1_2_2_2"/><b>then </b><i><font color="Green" title="E53">A26</font></i>: 
<font color="Maroon" title="c13">FI</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a>
 <b>by </b><i><a class="ref" href="vectsp11.html#T16" title="VECTSP11:th.16">VECTSP11:16</a></i>;<br/><a NAME="E3:53_2_1_2_2_2"/>
( <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="#E1:53_2_1_2"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="#E9:53_2_1_2"><i><font color="Green" title="E47">A11</font></i></a>, <a class="txt" href="#E16:53_2_1_2"><i><font color="Green" title="E51">A15</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E4:53_2_1_2_2_2"/><b>then </b>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:53_2_1_2_2_2"/><b>then </b>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">Ji</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font>, <font color="Maroon" title="c17">Bim</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span><b> such that </b><br/><a NAME="E7:53_2_1_2_2_2"/><i><font color="Green" title="E54">A27</font></i>: 
 <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c13">FI</font>,<font color="Maroon" title="c17">Bim</font>,<font color="Maroon" title="c17">Bim</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c16">Ji</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
 <b>and </b><br/><a NAME="E8:53_2_1_2_2_2"/><i><font color="Green" title="E55">A28</font></i>: 
for <font color="Olive" title="b1">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b1">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c13">FI</font> iff  ex <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b2">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="c16">Ji</font> &amp; <font color="Maroon" title="c16">Ji</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b1">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">Ji</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b2">i</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="#E1:53_2"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="#E1:53_2_1_2_2_2"><i><font color="Green" title="E52">A25</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c18">JJ</font> = <font color="Maroon" title="c14">Jk</font> <a href="matrixj2.html#K3" title="MATRIXJ2:func.3">^</a> <font color="Maroon" title="c16">Ji</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_1_2_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">x</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="c14">Jk</font> <a href="matrixj2.html#K3" title="MATRIXJ2:func.3">^</a> <font color="Maroon" title="c16">Ji</font></span>)</span> implies  not <span class="p1">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj2.html#K3" title="MATRIXJ2:func.3">^</a> <font color="Maroon" title="c16">Ji</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><br/><b>assume </b><a NAME="E1:53_2_1_2_2_2_1"/><i><font color="Green" title="E56">A29</font></i>: 
<font color="Maroon" title="c19">x</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="c14">Jk</font> <a href="matrixj2.html#K3" title="MATRIXJ2:func.3">^</a> <font color="Maroon" title="c16">Ji</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <span class="p1">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj2.html#K3" title="MATRIXJ2:func.3">^</a> <font color="Maroon" title="c16">Ji</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/><b>reconsider </b><font color="Maroon" title="c20">i</font> = <font color="Maroon" title="c19">x</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="#E1:53_2_1_2_2_2_1"><i><font color="Green" title="E56">A29</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_1_2_2_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_2_2_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_2_1_2_2_2_1_1_1"/>
( <font color="Maroon" title="c20">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="c14">Jk</font> or  ex <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">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="c16">Ji</font> &amp; <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Jk</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font> ) )
 </span><b>by </b><i><a class="txt" href="#E1:53_2_1_2_2_2_1"><i><font color="Green" title="E56">A29</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_2_2_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:53_2_1_2_2_2_1_1_1_1"/>
<font color="Maroon" title="c20">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="c14">Jk</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <span class="p1">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj2.html#K3" title="MATRIXJ2:func.3">^</a> <font color="Maroon" title="c16">Ji</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c20">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/><div class="add"><a NAME="E2:53_2_1_2_2_2_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj2.html#K3" title="MATRIXJ2:func.3">^</a> <font color="Maroon" title="c16">Ji</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c20">i</font> &amp; not <font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c20">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>;<br/><b>hence </b><a NAME="E3:53_2_1_2_2_2_1_1_1_1"/>
not <span class="p1">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj2.html#K3" title="MATRIXJ2:func.3">^</a> <font color="Maroon" title="c16">Ji</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c20">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_2_2_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:53_2_1_2_2_2_1_1_1_2"/>
 ex <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">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="c16">Ji</font> &amp; <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Jk</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <span class="p1">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj2.html#K3" title="MATRIXJ2:func.3">^</a> <font color="Maroon" title="c16">Ji</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c20">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c21">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:53_2_1_2_2_2_1_1_1_2"/><i><font color="Green" title="E57">A30</font></i>: 
( <font color="Maroon" title="c21">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="c16">Ji</font> &amp; <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Jk</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">j</font> )
 ;<br/><a NAME="E4:53_2_1_2_2_2_1_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj2.html#K3" title="MATRIXJ2:func.3">^</a> <font color="Maroon" title="c16">Ji</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Ji</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c21">j</font> &amp; not <font color="Maroon" title="c16">Ji</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c21">j</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="#E3:53_2_1_2_2_2_1_1_1_2"><i><font color="Green" title="E57">A30</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>;<br/><b>hence </b><a NAME="E5:53_2_1_2_2_2_1_1_1_2"/>
not <span class="p1">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj2.html#K3" title="MATRIXJ2:func.3">^</a> <font color="Maroon" title="c16">Ji</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c20">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:53_2_1_2_2_2_1"/>
not <span class="p1">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj2.html#K3" title="MATRIXJ2:func.3">^</a> <font color="Maroon" title="c16">Ji</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c19">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c19">JJ</font> = <font color="Maroon" title="c14">Jk</font> <a href="matrixj2.html#K3" title="MATRIXJ2:func.3">^</a> <font color="Maroon" title="c16">Ji</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font> <b>by </b><i><a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>;<br/><a NAME="E11:53_2_1_2_2_2"/>
<font color="Maroon" title="c15">Bker</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c17">Bim</font> is    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E15:53_2_1_2"><i><font color="Green" title="E50">A14</font></i></a>, <a class="ref" href="matrlin2.html#T26" title="MATRLIN2:th.26">MATRLIN2:26</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c20">BB</font> = <font color="Maroon" title="c15">Bker</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c17">Bim</font> as    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">V</font> <b>by </b><i><a class="txt" href="#E16:53_2_1_2"><i><font color="Green" title="E51">A15</font></i></a>, <a class="ref" href="matrlin2.html#T4" title="MATRLIN2:th.4">MATRLIN2:4</a></i>;<br/><b>take </b>
<font color="Maroon" title="c19">JJ</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c5">F</font>,<font color="Olive" title="b1">b1</font>,<font color="Olive" title="b1">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c19">JJ</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b2">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b2">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font> iff  ex <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b3">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="c19">JJ</font> &amp; <font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b2">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b3">i</font></span>)</span></span>)</span> ) ) ) )</span><br/><b>take </b>
<font color="Maroon" title="c20">BB</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c20">BB</font>,<font color="Maroon" title="c20">BB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c19">JJ</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b1">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b1">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font> iff  ex <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b2">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="c19">JJ</font> &amp; <font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b1">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b2">i</font></span>)</span></span>)</span> ) ) ) )</span><br/><a NAME="E13:53_2_1_2_2_2"/>
( (  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) &amp; (  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) )
 ;<br/><b>hence </b> <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c20">BB</font>,<font color="Maroon" title="c20">BB</font> = 
 <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <span class="p1"><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c14">Jk</font>,<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c16">Ji</font>,<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span></span><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
<b>by </b><i><a class="txt" href="#E13:53_2_1_2"><i><font color="Green" title="E48">A12</font></i></a>, <a class="txt" href="#E7:53_2_1_2_2_2"><i><font color="Green" title="E54">A27</font></i></a>, <a class="txt" href="#E15:53_2_1_2"><i><font color="Green" title="E50">A14</font></i></a>, <a class="ref" href="matrlin2.html#T27" title="MATRLIN2:th.27">MATRLIN2:27</a></i>
<br/>.= 
 <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrixj1.html#K16" title="MATRIXJ1:func.16">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c14">Jk</font>,<span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span></span><a href="matrixj1.html#K16" title="MATRIXJ1:func.16">*&gt;</a></span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c16">Ji</font></span>)</span>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixj1.html#T36" title="MATRIXJ1:th.36">MATRIXJ1:36</a></i>
<br/>.= 
 <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c19">JJ</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixj1.html#T35" title="MATRIXJ1:th.35">MATRIXJ1:35</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b1">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font> iff  ex <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b2">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="c19">JJ</font> &amp; <font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b1">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b2">i</font></span>)</span></span>)</span> ) )</span><br/><b>let </b><font color="Maroon" title="c21">L1</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c21">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font> iff  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <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="c19">JJ</font> &amp; <font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c21">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> ) )</span><br/><b>thus </b><a NAME="E15:53_2_1_2_2_2"/>
( <font color="Maroon" title="c21">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font> implies  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <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="c19">JJ</font> &amp; <font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c21">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <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="c19">JJ</font> &amp; <font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c21">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> ) implies <font color="Maroon" title="c21">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_1_2_2_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_2_1_2_2_2_3"/><i><font color="Green" title="E56">A31</font></i>: 
<font color="Maroon" title="c21">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <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="c19">JJ</font> &amp; <font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c21">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_2_2_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_2_1_2_2_2_3_1"/>
( <font color="Maroon" title="c7">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">L1</font> or <font color="Maroon" title="c7">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">L1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_2_2_2_3_1_1"><b>suppose </b></a><a NAME="E1:53_2_1_2_2_2_3_1_1"/><i><font color="Green" title="E57">A32</font></i>: 
<font color="Maroon" title="c7">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">L1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <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="c19">JJ</font> &amp; <font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c21">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> )</span><br/><div class="add"><b>take </b><font color="Maroon" title="c22">i</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Jk</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c22">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="c19">JJ</font> &amp; <font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c21">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font></span>)</span></span>)</span> )</span><br/><a NAME="E2:53_2_1_2_2_2_3_1_1"/>
<font color="Maroon" title="c22">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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Jk</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E14:53_2_1_2"><i><font color="Green" title="E49">A13</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/><a NAME="E3:53_2_1_2_2_2_3_1_1"/><b>then </b><i><font color="Green" title="E58">A33</font></i>: 
<font color="Maroon" title="c22">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="c14">Jk</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:53_2_1_2_2_2_3_1_1"/><i><font color="Green" title="E59">A34</font></i>: 
<font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c7">L</font>,<font color="Maroon" title="c23">k</font>
 <b>by </b><i><a class="ref" href="matrixj2.html#D3" target="_self" title="MATRIXJ2:def.3">Def3</a></i>;<br/><a NAME="E6:53_2_1_2_2_2_3_1_1"/>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">Jk</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">JJ</font> &amp; <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font></span>)</span> &amp; <font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font> )
 <b>by </b><i><a class="txt" href="#E3:53_2_1_2_2_2_3_1_1"><i><font color="Green" title="E58">A33</font></i></a>, <a class="txt" href="#E5:53_2_1_2_2_2_3_1_1"><i><font color="Green" title="E59">A34</font></i></a>, <a class="ref" href="matrixj2.html#D1" target="_self" title="MATRIXJ2:def.1">Def1</a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E7:53_2_1_2_2_2_3_1_1"/>
( <font color="Maroon" title="c22">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="c19">JJ</font> &amp; <font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c21">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:53_2_1_2_2_2_3_1_1"><i><font color="Green" title="E57">A32</font></i></a>, <a class="txt" href="#E3:53_2_1_2_2_2_3_1_1"><i><font color="Green" title="E58">A33</font></i></a>, <a class="txt" href="#E5:53_2_1_2_2_2_3_1_1"><i><font color="Green" title="E59">A34</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_2_2_2_3_1_2"><b>suppose </b></a><a NAME="E1:53_2_1_2_2_2_3_1_2"/>
<font color="Maroon" title="c7">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">L1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <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="c19">JJ</font> &amp; <font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c21">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> )</span><br/><div class="add"><a NAME="E2:53_2_1_2_2_2_3_1_2"/><b>then </b>
<font color="Maroon" title="c21">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c13">FI</font>
 <b>by </b><i><a class="txt" href="#E2:53_2_1_2"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="#E7:53_2_1_2"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="#E6:53_2_1_2"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="#E16:53_2_1_2"><i><font color="Green" title="E51">A15</font></i></a>, <a class="txt" href="#E1:53_2_1_2_2_2_3"><i><font color="Green" title="E56">A31</font></i></a>, <a class="ref" href="vectsp11.html#T39" title="VECTSP11:th.39">VECTSP11:39</a></i>;<br/><b>then consider </b><font color="Maroon" title="c22">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:53_2_1_2_2_2_3_1_2"/><i><font color="Green" title="E57">A35</font></i>: 
( <font color="Maroon" title="c22">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="c16">Ji</font> &amp; <font color="Maroon" title="c16">Ji</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c21">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">Ji</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E8:53_2_1_2_2_2"><i><font color="Green" title="E55">A28</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c23">ii</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Jk</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c23">ii</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="c19">JJ</font> &amp; <font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c23">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c21">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c23">ii</font></span>)</span></span>)</span> )</span><br/><a NAME="E5:53_2_1_2_2_2_3_1_2"/>
( <font color="Maroon" title="c23">ii</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="c19">JJ</font> &amp; <font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c23">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Ji</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font> )
 <b>by </b><i><a class="txt" href="#E4:53_2_1_2_2_2_3_1_2"><i><font color="Green" title="E57">A35</font></i></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E6:53_2_1_2_2_2_3_1_2"/>
( <font color="Maroon" title="c23">ii</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="c19">JJ</font> &amp; <font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c23">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c21">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c23">ii</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:53_2_1_2_2_2_3_1_2"><i><font color="Green" title="E57">A35</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>given </b><font color="Maroon" title="c22">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E17:53_2_1_2_2_2"/><i><font color="Green" title="E56">A36</font></i>: 
( <font color="Maroon" title="c22">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="c19">JJ</font> &amp; <font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c21">L1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font></span>)</span></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c21">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_2_2_2_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_2_1_2_2_2_4"/>
( <font color="Maroon" title="c22">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="c14">Jk</font> or  ex <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">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="c16">Ji</font> &amp; <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Jk</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font> ) )
 </span><b>by </b><i><a class="txt" href="#E17:53_2_1_2_2_2"><i><font color="Green" title="E56">A36</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_2_2_2_4_1"><b>suppose </b></a><a NAME="E1:53_2_1_2_2_2_4_1"/><i><font color="Green" title="E57">A37</font></i>: 
<font color="Maroon" title="c22">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="c14">Jk</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c21">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font></span><br/><div class="add"><a NAME="E2:53_2_1_2_2_2_4_1"/><b>then </b><i><font color="Green" title="E58">A38</font></i>: 
<font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>consider </b><font color="Maroon" title="c23">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:53_2_1_2_2_2_4_1"/><i><font color="Green" title="E59">A39</font></i>: 
<font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c7">L</font>,<font color="Maroon" title="c23">k</font>
 <b>by </b><i><a class="txt" href="#E1:53_2_1_2_2_2_4_1"><i><font color="Green" title="E57">A37</font></i></a>, <a class="ref" href="matrixj2.html#D3" target="_self" title="MATRIXJ2:def.3">Def3</a></i>;<br/><a NAME="E5:53_2_1_2_2_2_4_1"/>
<font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E1:53_2_1_2_2_2_4_1"><i><font color="Green" title="E57">A37</font></i></a>, <a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>;<br/><a NAME="E6:53_2_1_2_2_2_4_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font></span>)</span> <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/><a NAME="E7:53_2_1_2_2_2_4_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:53_2_1_2_2_2_4_1"/><b>then </b><i><font color="Green" title="E60">A40</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/><b>then </b><font color="Maroon" title="c7">L</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">Jk</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:53_2_1_2_2_2_4_1"><i><font color="Green" title="E59">A39</font></i></a>, <a class="ref" href="matrixj2.html#D1" target="_self" title="MATRIXJ2:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c21">L1</font>
<b>by </b><i><a class="txt" href="#E2:53_2_1_2_2_2_4_1"><i><font color="Green" title="E58">A38</font></i></a>, <a class="txt" href="#E8:53_2_1_2_2_2_4_1"><i><font color="Green" title="E60">A40</font></i></a>, <a class="txt" href="#E17:53_2_1_2_2_2"><i><font color="Green" title="E56">A36</font></i></a>, <a class="ref" href="matrixj2.html#D1" target="_self" title="MATRIXJ2:def.1">Def1</a></i>
;<br/><b>hence </b><a NAME="E10:53_2_1_2_2_2_4_1"/>
<font color="Maroon" title="c21">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="#E2:53_2_1_2"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="#E4:53_2_1_2"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="vectsp11.html#D2" title="VECTSP11:def.2">VECTSP11:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_2_2_2_4_2"><b>suppose </b></a><a NAME="E1:53_2_1_2_2_2_4_2"/>
 ex <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">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="c16">Ji</font> &amp; <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Jk</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c21">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c23">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:53_2_1_2_2_2_4_2"/><i><font color="Green" title="E57">A41</font></i>: 
( <font color="Maroon" title="c23">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="c16">Ji</font> &amp; <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Jk</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font> )
 ;<br/><a NAME="E4:53_2_1_2_2_2_4_2"/>
<font color="Maroon" title="c19">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Ji</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c23">j</font>
 <b>by </b><i><a class="txt" href="#E3:53_2_1_2_2_2_4_2"><i><font color="Green" title="E57">A41</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E5:53_2_1_2_2_2_4_2"/><b>then </b>
<font color="Maroon" title="c21">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c13">FI</font>
 <b>by </b><i><a class="txt" href="#E17:53_2_1_2_2_2"><i><font color="Green" title="E56">A36</font></i></a>, <a class="txt" href="#E3:53_2_1_2_2_2_4_2"><i><font color="Green" title="E57">A41</font></i></a>, <a class="txt" href="#E8:53_2_1_2_2_2"><i><font color="Green" title="E55">A28</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">w</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:53_2_1_2_2_2_4_2"/><i><font color="Green" title="E58">A42</font></i>: 
( <font color="Maroon" title="c24">w</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> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">FI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">L1</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c24">w</font> )
 <b>by </b><i><a class="txt" href="#E2:53_2_1_2_2_2"><i><font color="Green" title="E53">A26</font></i></a>, <a class="ref" href="vectsp11.html#D2" title="VECTSP11:def.2">VECTSP11:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">W</font> = <font color="Maroon" title="c24">w</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">V</font> <b>by </b><i><a class="ref" href="vectsp_4.html#T18" title="VECTSP_4:th.18">VECTSP_4:18</a></i>;<br/><a NAME="E9:53_2_1_2_2_2_4_2"/><i><font color="Green" title="E59">A43</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <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="c4">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T19" title="VECTSP_4:th.19">VECTSP_4:19</a></i>;<br/><font color="Maroon" title="c21">L1</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c25">W</font> = 
<font color="Maroon" title="c13">FI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">w</font>
<b>by </b><i><a class="txt" href="#E7:53_2_1_2_2_2_4_2"><i><font color="Green" title="E58">A42</font></i></a>, <a class="ref" href="vectsp_4.html#T22" title="VECTSP_4:th.22">VECTSP_4:22</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">W</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><b>hence </b><a NAME="E11:53_2_1_2_2_2_4_2"/>
<font color="Maroon" title="c21">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="#E2:53_2_1_2"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="#E7:53_2_1_2_2_2_4_2"><i><font color="Green" title="E58">A42</font></i></a>, <a class="txt" href="#E9:53_2_1_2_2_2_4_2"><i><font color="Green" title="E59">A43</font></i></a>, <a class="ref" href="vectsp11.html#D2" title="VECTSP11:def.2">VECTSP11:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:53"/><i><font color="Green" title="E40">A44</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E1:53"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="#E2:53"><i><font color="Green" title="E39">A3</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c2">V</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c2">V</font>  ex <font color="Olive" title="b2">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font> ex <font color="Olive" title="b3">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c2">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">b1</font>,<font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b2">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b4">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b4">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b1">F</font> iff  ex <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b5">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="Olive" title="b2">J</font> &amp; <font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b4">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font></span>)</span></span>)</span> ) ) ) )</span><br/><b>let </b><font color="Maroon" title="c3">F</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font> ex <font color="Olive" title="b2">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c2">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c3">F</font>,<font color="Olive" title="b2">b1</font>,<font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b1">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b3">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b3">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c3">F</font> iff  ex <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b4">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="Olive" title="b1">J</font> &amp; <font color="Olive" title="b1">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b3">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> ) ) ) )</span><br/>



<a NAME="E4:53"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font>]
 
<b>by </b><i><a class="txt" href="#E3:53"><i><font color="Green" title="E40">A44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:53"/>
 ex <font color="Olive" title="b1">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="matrixj2.html#NM1" title="MATRIXJ2:NM.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c1">K</font> ex <font color="Olive" title="b2">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c2">V</font> st <br/>(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c3">F</font>,<font color="Olive" title="b2">b1</font>,<font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b1">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; ( for <font color="Olive" title="b3">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Olive" title="b3">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c3">F</font> iff  ex <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b4">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="Olive" title="b1">J</font> &amp; <font color="Olive" title="b1">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Olive" title="b3">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> ) ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
