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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">V</font> being  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Olive" title="b1">K</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b2">V</font>,<font color="Olive" title="b2">V</font> holds  <font color="Olive" title="b3">f</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> is  <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">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  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c2">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">V</font>,<font color="Olive" title="b1">V</font> holds  <font color="Olive" title="b2">f</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> is  <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">V</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c2">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="c3">V</font>,<font color="Maroon" title="c3">V</font> holds  <font color="Olive" title="b1">f</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> is  <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c3">V</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">f</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> is  <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><br/>







<b>set </b><font color="Maroon" title="c5">KER</font> =  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">fK</font> = <font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> as   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="vectsp11.html#T28" title="VECTSP11:th.28">VECTSP11:28</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">v</font> be   <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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">fK</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <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>  <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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c8">v1</font> = <font color="Maroon" title="c7">v</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">V</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="E2:32_1"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c8">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">fK</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
<b>by </b><i><a class="ref" href="vectsp11.html#T22" title="VECTSP11:th.22">VECTSP11:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">v1</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">V</font>
<b>by </b><i><a class="txt" href="#E2:32_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="ranknull.html#T10" title="RANKNULL:th.10">RANKNULL:10</a></i>
<br/>.= 
 <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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_4.html#T19" title="VECTSP_4:th.19">VECTSP_4:19</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>hence </b><a NAME="E3:32"/>
<font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> is  <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="matrixj2.html#D4" target="_self" title="MATRIXJ2:def.4">Def4</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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