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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#NM9" title="VECTSP_1:NM.9">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#NM8" title="VECTSP_1:NM.8">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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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/><span class="kw">let </span><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#NM9" title="VECTSP_1:NM.9">VectSp</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">w</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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/><span class="kw">let </span><font color="Maroon" title="c3">w</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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/>







<span class="kw">assume </span><a NAME="E1:16"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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/>

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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/>



<span class="kw">consider </span><font color="Maroon" title="c7">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E3:16"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:16"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<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#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c7">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#D2" target="_self" onmouseover="rs('vectsp11/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="lab"><font color="Green" title="E16">A4</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c7">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="matrlin.html#D4" onmouseover="rs('matrlin/D4')" onmouseout="rh()">MATRLIN:def 4</a></span>
<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#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c7">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" onmouseover="rs('vectsp_1/D16')" onmouseout="rh()">VECTSP_1:def 16</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:16"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <font color="Maroon" title="c5">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="txt" href="#E3:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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/>

<span class="kw">hence </span><a NAME="E7:16"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
<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#K4" title="MATRLIN:func.4">*</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="vectsp11.html#D2" target="_self" onmouseover="rs('vectsp11/D2')" onmouseout="rh()">Def2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K4" title="MATRLIN:func.4">*</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"><span class="kw">assume </span><a NAME="E1:16_2"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K4" title="MATRLIN:func.4">*</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#K4" title="MATRLIN:func.4">*</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#K3" title="VECTSP_1:func.3">*</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="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D4" onmouseover="rs('matrlin/D4')" onmouseout="rh()">MATRLIN:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c3">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_2"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="vectsp11.html#D3" target="_self" onmouseover="rs('vectsp11/D3')" onmouseout="rh()">Def3</a></span>
<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#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c3">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" onmouseover="rs('vectsp_1/D16')" onmouseout="rh()">VECTSP_1:def 16</a></span>
;<br/><span class="kw">hence </span><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#K4" title="MATRLIN:func.4">*</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="ref" href="vectsp11.html#D3" target="_self" onmouseover="rs('vectsp11/D3')" onmouseout="rh()">Def3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E9:16"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
<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#K4" title="MATRLIN:func.4">*</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K3" title="VECTSP_1:func.3">*</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#K4" title="MATRLIN:func.4">*</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D4" onmouseover="rs('matrlin/D4')" onmouseout="rh()">MATRLIN:def 4</a></span>
<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#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c3">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="vectsp11.html#D3" target="_self" onmouseover="rs('vectsp11/D3')" onmouseout="rh()">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c3">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" onmouseover="rs('vectsp_1/D16')" onmouseout="rh()">VECTSP_1:def 16</a></span>
;<br/>
<span class="kw">then </span> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</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#K3" title="VECTSP_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T16" onmouseover="rs('vectsp_1/T16')" onmouseout="rh()">VECTSP_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</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#K3" title="VECTSP_1:func.3">*</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#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T22" onmouseover="rs('vectsp_1/T22')" onmouseout="rh()">VECTSP_1:22</a></span>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</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="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D14" onmouseover="rs('vectsp_1/D14')" onmouseout="rh()">VECTSP_1:def 14</a></span>
;<br/>
<a NAME="E12:16"/><span class="kw">then </span>
<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="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="vectsp_1.html#T15" onmouseover="rs('vectsp_1/T15')" onmouseout="rh()">VECTSP_1:15</a></span>;<br/>
<a NAME="E13:16"/><span class="kw">then </span>
 <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#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c3">w</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T16" onmouseover="rs('vectsp_1/T16')" onmouseout="rh()">VECTSP_1:16</a></span>;<br/>
<a NAME="E14:16"/><span class="kw">then </span>
<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#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c3">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T18" onmouseover="rs('rlvect_1/T18')" onmouseout="rh()">RLVECT_1:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:16"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#D3" target="_self" onmouseover="rs('vectsp11/D3')" onmouseout="rh()">Def3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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