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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">V1</font> being  <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>  st not  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Olive" title="b2">f</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> holds <br/>( <font color="Olive" title="b2">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b2">f</font> )</span><br/><b>let </b><font color="Maroon" title="c2">V1</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c2">V1</font>  st not  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Olive" title="b1">f</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> holds <br/>( <font color="Olive" title="b1">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b1">f</font> )</span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c2">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c3">f</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> implies ( <font color="Maroon" title="c3">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c3">f</font> ) )</span><br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E15">A1</font></i>: 
not  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c3">f</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c3">f</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c4">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c3">f</font></span>)</span><b> such that </b><br/><a NAME="E3:20"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:20"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="anproj_1.html#D8" title="ANPROJ_1:def.8">ANPROJ_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">v</font> = <font color="Maroon" title="c4">v</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V1</font> <b>by </b><i><a class="ref" href="vectsp_4.html#T18" title="VECTSP_4:th.18">VECTSP_4:18</a></i>;<br/>
<a NAME="E5:20"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c5">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">V1</font>
 
<b>by </b><i><a class="txt" href="#E3:20"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="vectsp_4.html#T19" title="VECTSP_4:th.19">VECTSP_4:19</a></i>;<br/>
<i><font color="Green" title="E18">A4</font></i>: <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font>
<b>by </b><i><a class="ref" href="ranknull.html#T14" title="RANKNULL:th.14">RANKNULL:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T59" title="VECTSP_1:th.59">VECTSP_1:59</a></i>
;<br/>
<a NAME="E7:20"/><b>then </b>
<font color="Maroon" title="c3">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a>
 
<b>by </b><i><a class="txt" href="#E5:20"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="vectsp11.html#D1" target="_self" title="VECTSP11:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E8:20"/>
( <font color="Maroon" title="c3">f</font> is <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c3">f</font> )
 <b>by </b><i><a class="txt" href="#E5:20"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E6:20"><i><font color="Green" title="E18">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"> verum</span><br/>


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