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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">V</font> being non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</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">w</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b1">V</font><br/> for <font color="Olive" title="b3">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b3">a</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="c1">K</font> holds <br/>for <font color="Olive" title="b4">f</font> being <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">V</font>,<font color="Olive" title="b1">V</font><br/> for <font color="Olive" title="b5">L</font> being   <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b4">f</font> holds <br/> ( <font color="Olive" title="b3">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b4">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp; <font color="Olive" title="b3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b5">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b3">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b4">f</font> &amp; ( <font color="Olive" title="b2">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Olive" title="b4">f</font>,<font color="Olive" title="b5">L</font> implies <font color="Olive" title="b2">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Olive" title="b3">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b4">f</font>,<font color="Olive" title="b3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b5">L</font> ) &amp; ( <font color="Olive" title="b2">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Olive" title="b3">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b4">f</font>,<font color="Olive" title="b3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b5">L</font> implies <font color="Olive" title="b2">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Olive" title="b4">f</font>,<font color="Olive" title="b5">L</font> ) )</span><br/><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="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">w</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b2">a</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="c1">K</font> holds <br/>for <font color="Olive" title="b3">f</font> being <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b4">L</font> being   <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b3">f</font> holds <br/> ( <font color="Olive" title="b2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b3">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp; <font color="Olive" title="b2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b4">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b3">f</font> &amp; ( <font color="Olive" title="b1">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">L</font> implies <font color="Olive" title="b1">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Olive" title="b2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b4">L</font> ) &amp; ( <font color="Olive" title="b1">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Olive" title="b2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b4">L</font> implies <font color="Olive" title="b1">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">L</font> ) )</span><br/><b>let </b><font color="Maroon" title="c3">w</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <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"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">a</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="c1">K</font> holds <br/>for <font color="Olive" title="b2">f</font> being <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b3">L</font> being   <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b2">f</font> holds <br/> ( <font color="Olive" title="b1">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b2">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp; <font color="Olive" title="b1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b3">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b1">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b2">f</font> &amp; ( <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">L</font> implies <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Olive" title="b1">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b3">L</font> ) &amp; ( <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Olive" title="b1">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b3">L</font> implies <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">L</font> ) )</span><br/><b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c4">a</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="c1">K</font> implies for <font color="Olive" title="b1">f</font> being <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b2">L</font> being   <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b1">f</font> holds <br/> ( <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <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="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b1">f</font> &amp; ( <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">L</font> implies <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">L</font> ) &amp; ( <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">L</font> implies <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">L</font> ) ) )</span><br/>







<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c4">a</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="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="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b2">L</font> being   <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b1">f</font> holds <br/> ( <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <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="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b1">f</font> &amp; ( <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">L</font> implies <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">L</font> ) &amp; ( <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">L</font> implies <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">L</font> ) )</span><br/>

<b>let </b><font color="Maroon" title="c5">f</font> be  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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"> for <font color="Olive" title="b1">L</font> being   <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c5">f</font> holds <br/> ( <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp; <font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font> &amp; ( <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c5">f</font>,<font color="Olive" title="b1">L</font> implies <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">L</font> ) &amp; ( <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">L</font> implies <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c5">f</font>,<font color="Olive" title="b1">L</font> ) )</span><br/><b>let </b><font color="Maroon" title="c6">L</font> be    <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"> ( <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp; <font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font> &amp; ( <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">L</font> implies <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">L</font> ) &amp; ( <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">L</font> implies <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">L</font> ) )</span><br/>



<b>consider </b><font color="Maroon" title="c7">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:16"/><i><font color="Green" title="E15">A2</font></i>: 
( <font color="Maroon" title="c7">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="c2">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="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">v</font> )
 <b>by </b><i><a class="ref" href="vectsp11.html#D2" target="_self" title="VECTSP11:def.2">Def2</a></i>;<br/>
<i><font color="Green" title="E16">A3</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> = 
<font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">v</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="matrlin.html#D6" title="MATRLIN:def.6">MATRLIN:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">v</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
;<br/>
<b>hence </b><a NAME="E5:16"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <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="txt" href="#E3:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="vectsp11.html#D1" target="_self" title="VECTSP11:def.1">Def1</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="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font> &amp; ( <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">L</font> implies <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">L</font> ) &amp; ( <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">L</font> implies <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">L</font> ) )</span><br/>

<b>thus </b><a NAME="E6:16"/><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">L</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="#E3:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="#E4:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E5:16"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="vectsp11.html#D2" target="_self" title="VECTSP11:def.2">Def2</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="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">L</font> implies <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">L</font> ) &amp; ( <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">L</font> implies <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">L</font> ) )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">L</font> implies <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">L</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:16_2"/><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">L</font></span><br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">w</font> = 
<font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">w</font></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#D6" title="MATRLIN:def.6">MATRLIN:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c3">w</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:16_2"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="vectsp11.html#D3" target="_self" title="VECTSP11:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c3">w</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
;<br/><b>hence </b><a NAME="E3:16_2"/>
<font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">L</font>
 <b>by </b><i><a class="txt" href="#E5:16"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="#E6:16"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="vectsp11.html#D3" target="_self" title="VECTSP11:def.3">Def3</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="E8:16"/><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">L</font></span><br/>

<font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">w</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">w</font>
<b>by </b><i><a class="ref" href="matrlin.html#D6" title="MATRLIN:def.6">MATRLIN:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c3">w</font>
<b>by </b><i><a class="txt" href="#E5:16"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="#E6:16"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="#E8:16"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="vectsp11.html#D3" target="_self" title="VECTSP11:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c3">w</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
;<br/>
<b>then </b> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T63" title="VECTSP_1:th.63">VECTSP_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T69" title="VECTSP_1:th.69">VECTSP_1:69</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
;<br/>
<a NAME="E11:16"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c3">w</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="c2">V</font>
 
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="vectsp_1.html#T60" title="VECTSP_1:th.60">VECTSP_1:60</a></i>;<br/>
<a NAME="E12:16"/><b>then </b>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c3">w</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#T63" title="VECTSP_1:th.63">VECTSP_1:63</a></i>;<br/>
<a NAME="E13:16"/><b>then </b>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c3">w</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T31" title="RLVECT_1:th.31">RLVECT_1:31</a></i>;<br/>
<b>hence </b><a NAME="E14:16"/>
<font color="Maroon" title="c3">w</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">L</font>
 <b>by </b><i><a class="ref" href="vectsp11.html#D3" target="_self" title="VECTSP11:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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