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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V1</font> being non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="matrlin.html#V1" title="MATRLIN:attr.1">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">V1</font>,<font color="Olive" title="b1">V1</font> holds  <font color="Olive" title="b2">f</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">V1</font> be  non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="matrlin.html#V1" title="MATRLIN:attr.1">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">V1</font>,<font color="Maroon" title="c2">V1</font> holds  <font color="Olive" title="b1">f</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> </span><br/><span class="kw">let </span><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">V1</font>,<font color="Maroon" title="c2">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> </span><br/>





<span class="kw">consider </span><font color="Maroon" title="c4">b1</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c2">V1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">AutA</font> =  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">P</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">K</font><span class="kw"> such that </span><br/><a NAME="E3:23"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c6">P</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="c4">b1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E4:23"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> holds   <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c6">P</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T8" target="_self" title="VECTSP11:th.8">Th8</a></span>;<br/>
<a NAME="E5:23"/>
(  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V1</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>  &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T21" title="MATRLIN2:th.21">MATRLIN2:21</a>, <a class="ref" href="matrlin2.html#T42" title="MATRLIN2:th.42">MATRLIN2:42</a></span>;<br/>
<a NAME="E6:23"/><span class="kw">then </span>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c6">P</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<a NAME="E7:23"/><span class="kw">then </span>
<font color="Maroon" title="c6">P</font> is  <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D8" title="POLYNOM5:def.8">POLYNOM5:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">L</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><span class="kw"> such that </span><br/><a NAME="E9:23"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c7">L</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D7" title="POLYNOM5:def.7">POLYNOM5:def 7</a></span>;<br/>
<span class="lab"><font color="Green" title="E20">A4</font></span>:  <a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c2">V1</font></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font> = 
<font color="Maroon" title="c7">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c2">V1</font></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T38" title="MATRLIN2:th.38">MATRLIN2:38</a></span>
<br/>.= 
<font color="Maroon" title="c7">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c2">V1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T34" title="MATRLIN2:th.34">MATRLIN2:34</a></span>
<br/>.= 
 <a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">L</font> <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="c2">V1</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T34" title="MATRLIN2:th.34">MATRLIN2:34</a></span>
;<br/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> = 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">L</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="polynom5.html#D6" title="POLYNOM5:def.6">POLYNOM5:def 6</a></span>
<br/>.= 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>
<br/>.= 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c2">V1</font></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T28" title="MATRLIN2:th.28">MATRLIN2:28</a></span>
<br/>.= 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">+</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">L</font> <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="c2">V1</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="matrlin2.html#T39" title="MATRLIN2:th.39">MATRLIN2:39</a></span>
<br/>.= 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">L</font> <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="c2">V1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#T47" title="MATRLIN:th.47">MATRLIN:47</a></span>
<br/>.= 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">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> <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></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="c2">V1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T30" title="RLVECT_1:th.30">RLVECT_1:30</a></span>
<br/>.= 
 <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="c3">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> <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></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="c2">V1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#D2" title="MATRLIN2:def.2">MATRLIN2:def 2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:23"/>
<font color="Maroon" title="c3">f</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T15" target="_self" title="VECTSP11:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
