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<b>let </b><font color="Maroon" title="c1">K</font> be   <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">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">V1</font> being <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="b3">b1</font>, <font color="Olive" title="b4">b1'</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b2">V1</font><br/> for <font color="Olive" title="b5">f</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b2">V1</font>,<font color="Olive" title="b2">V1</font> holds <br/> ( ( <font color="Olive" title="b5">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp; <font color="Olive" title="b1">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b5">f</font> ) iff  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">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="Olive" title="b1">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="Olive" title="b2">V1</font></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b3">b1</font>,<font color="Olive" title="b4">b1'</font></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="c1">K</font> )</span><br/><b>let </b><font color="Maroon" title="c2">L</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"> for <font color="Olive" title="b1">V1</font> being <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">b1</font>, <font color="Olive" title="b3">b1'</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b1">V1</font><br/> for <font color="Olive" title="b4">f</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b1">V1</font>,<font color="Olive" title="b1">V1</font> holds <br/> ( ( <font color="Olive" title="b4">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp; <font color="Maroon" title="c2">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b4">f</font> ) iff  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">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="c2">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="Olive" title="b1">V1</font></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">b1'</font></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="c1">K</font> )</span><br/><b>let </b><font color="Maroon" title="c3">V1</font> be  <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">b1</font>, <font color="Olive" title="b2">b1'</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V1</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c3">V1</font> holds <br/> ( ( <font color="Olive" title="b3">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp; <font color="Maroon" title="c2">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b3">f</font> ) iff  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">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="c2">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="c3">V1</font></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">b1</font>,<font color="Olive" title="b2">b1'</font></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="c1">K</font> )</span><br/><b>let </b><font color="Maroon" title="c4">b1</font>, <font color="Maroon" title="c5">b1'</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V1</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="c3">V1</font>,<font color="Maroon" title="c3">V1</font> holds <br/> ( ( <font color="Olive" title="b1">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp; <font color="Maroon" title="c2">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b1">f</font> ) iff  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">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="c2">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="c3">V1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b1'</font></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="c1">K</font> )</span><br/><b>let </b><font color="Maroon" title="c6">f</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c3">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c6">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp; <font color="Maroon" title="c2">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c6">f</font> ) iff  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">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="c2">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="c3">V1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b1'</font></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="c1">K</font> )</span><br/>









<a NAME="E1:22"/><i><font color="Green" title="E17">A1</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">V1</font>
 
;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">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="c2">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="c3">V1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b1'</font></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="c1">K</font> implies ( <font color="Maroon" title="c6">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp; <font color="Maroon" title="c2">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c6">f</font> ) )</span>
<div class="add"><b>assume </b><a NAME="E1:22_1"/>
( <font color="Maroon" title="c6">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp; <font color="Maroon" title="c2">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c6">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">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="c2">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="c3">V1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b1'</font></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="c1">K</font></span><br/><a NAME="E2:22_1"/><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="c6">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="c2">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="c3">V1</font></span>)</span></span>)</span></span>)</span> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 <b>by </b><i><a class="ref" href="vectsp11.html#T14" target="_self" title="VECTSP11:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E3:22_1"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">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="c2">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="c3">V1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b1'</font></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="c1">K</font>
 <b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="matrlin2.html#T50" title="MATRLIN2:th.50">MATRLIN2:50</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>assume </b><a NAME="E3:22"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">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="c2">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="c3">V1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b1'</font></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="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="c6">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp; <font color="Maroon" title="c2">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c6">f</font> )</span><br/>

<a NAME="E4:22"/><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="c6">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="c2">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="c3">V1</font></span>)</span></span>)</span></span>)</span> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 
<b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="matrlin2.html#T50" title="MATRLIN2:th.50">MATRLIN2:50</a></i>;<br/>
<b>hence </b><a NAME="E5:22"/>
( <font color="Maroon" title="c6">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp; <font color="Maroon" title="c2">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c6">f</font> )
 <b>by </b><i><a class="ref" href="vectsp11.html#T14" target="_self" title="VECTSP11:th.14">Th14</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>
