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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">W</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b2">V</font><br/> for <font color="Olive" title="b4">f</font> being <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <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><br/> for <font color="Olive" title="b5">fI</font> being <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">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#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>  st <font color="Olive" title="b5">fI</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b4">f</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b4">f</font> holds <br/><span class="p1">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b5">fI</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b4">f</font></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">W</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b1">V</font><br/> for <font color="Olive" title="b3">f</font> being <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <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><br/> for <font color="Olive" title="b4">fI</font> being <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</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#K3" title="RANKNULL:func.3">im</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>  st <font color="Olive" title="b4">fI</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <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#K3" title="RANKNULL:func.3">im</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> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b3">f</font> holds <br/><span class="p1">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b4">fI</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b3">f</font></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">W</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">V</font><br/> for <font color="Olive" title="b2">f</font> being <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <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><br/> for <font color="Olive" title="b3">fI</font> being <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</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#K3" title="RANKNULL:func.3">im</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>  st <font color="Olive" title="b3">fI</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <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#K3" title="RANKNULL:func.3">im</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> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b2">f</font> holds <br/><span class="p1">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b3">fI</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b2">f</font></span><br/><b>let </b><font color="Maroon" title="c4">W</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <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"> for <font color="Olive" title="b1">f</font> being <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <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><br/> for <font color="Olive" title="b2">fI</font> being <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</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#K3" title="RANKNULL:func.3">im</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>  st <font color="Olive" title="b2">fI</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <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#K3" title="RANKNULL:func.3">im</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> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b2">fI</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c5">f</font> be  <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <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"> for <font color="Olive" title="b1">fI</font> being <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">fI</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c5">f</font> holds <br/><span class="p1">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b1">fI</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c5">f</font></span><br/>









<b>set </b><font color="Maroon" title="c6">IM</font> =  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c7">fI</font> be  <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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"> ( <font color="Maroon" title="c7">fI</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c5">f</font> implies <span class="p1">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">fI</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c5">f</font> )</span><br/>

<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c7">fI</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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"> (  not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c5">f</font> or <span class="p1">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">fI</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c5">f</font> )</span><br/>

<b>set </b><font color="Maroon" title="c8">D</font> =  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c5">f</font>;<br/>
<b>assume </b><a NAME="E2:38"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <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"> <span class="p1">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">fI</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c5">f</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c9">Dn</font> = <span class="p1">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/>
<a NAME="E4:38"/><i><font color="Green" title="E26">A2</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">fI</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c9">Dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">fI</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c9">Dn</font></span>)</span> implies  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">fI</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c9">Dn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> )</span><br/><b>assume </b><a NAME="E1:38_1"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">fI</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c9">Dn</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">fI</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c9">Dn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span><br/><b>reconsider </b><font color="Maroon" title="c11">X</font> = <font color="Maroon" title="c10">x</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="txt" href="#E1:38_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">v</font> = <font color="Maroon" title="c11">X</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="E4:38_1"/>
<font color="Maroon" title="c11">X</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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>then consider </b><font color="Maroon" title="c13">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font><b> such that </b><br/><a NAME="E6:38_1"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c13">w</font>
 <b>by </b><i><a class="ref" href="ranknull.html#T13" title="RANKNULL:th.13">RANKNULL:13</a></i>;<br/><a NAME="E7:38_1"/><i><font color="Green" title="E29">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <span class="p2">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">w</font> = 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">ZeroMap</a> <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c3">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">w</font>
<b>by </b><i><a class="ref" href="matrixj2.html#D5" target="_self" title="MATRIXJ2:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">V</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">w</font>
<b>by </b><i><a class="ref" href="grcat_1.html#D12" title="GRCAT_1:def.12">GRCAT_1:def 12</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="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><b>hence </b> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Dn</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">w</font>
<b>by </b><i><a class="ref" href="vectsp_4.html#T19" title="VECTSP_4:th.19">VECTSP_4:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">Dn</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">w</font>
<b>by </b><i><a class="ref" href="vectsp11.html#T20" title="VECTSP11:th.20">VECTSP11:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">Dn</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">v</font>
<b>by </b><i><a class="txt" href="#E6:38_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="#E7:38_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c9">Dn</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c11">X</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">fI</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c9">Dn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="#E1:38"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="vectsp11.html#T22" title="VECTSP11:th.22">VECTSP11:22</a></i>
;<br/> <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>then </b><font color="Maroon" title="c7">fI</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c9">Dn</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">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>
<b>by </b><i><a class="txt" href="#E4:38"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="funcop_1.html#T17" title="FUNCOP_1:th.17">FUNCOP_1:17</a></i>
<br/>.= 
 <a href="grcat_1.html#K7" title="GRCAT_1:func.7">ZeroMap</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="grcat_1.html#D12" title="GRCAT_1:def.12">GRCAT_1:def 12</a></i>
;<br/>
<a NAME="E7:38"/><b>then </b><i><font color="Green" title="E27">A6</font></i>: 
 <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">fI</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">Dn</font>
 
<b>by </b><i><a class="ref" href="matrixj2.html#D5" target="_self" title="MATRIXJ2:def.5">Def5</a></i>;<br/>
<a NAME="E8:38"/>
 <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">fI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Dn</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c10">DI</font> =  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">fI</font>;<br/>
<b>assume </b><a NAME="E1:38_3"/>
 <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">fI</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">Dn</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:38_3"/><b>then </b>
 <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">fI</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">Dn</font>
 
<b>by </b><i><a class="txt" href="#E7:38"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E3:38_3"/><b>then </b><i><font color="Green" title="E28">A7</font></i>: 
<span class="p1">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">fI</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">Dn</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">V</font><b> such that </b><br/><a NAME="E5:38_3"/><i><font color="Green" title="E29">A8</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c5">f</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">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="c3">V</font>
 <b>by </b><i><a class="ref" href="matrixj2.html#T16" target="_self" title="MATRIXJ2:th.16">Th16</a></i>;<br/>
<a NAME="E6:38_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="ranknull.html#T13" title="RANKNULL:th.13">RANKNULL:13</a></i>;<br/>
<a NAME="E7:38_3"/><b>then </b><i><font color="Green" title="E30">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E8:38_3"/><i><font color="Green" title="E31">A10</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<font color="Maroon" title="c7">fI</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p1">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">fI</font></span>)</span> = 
 <a href="grcat_1.html#K7" title="GRCAT_1:func.7">ZeroMap</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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#D5" target="_self" title="MATRIXJ2:def.5">Def5</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">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>
<b>by </b><i><a class="ref" href="grcat_1.html#D12" title="GRCAT_1:def.12">GRCAT_1:def 12</a></i>
;<br/>
<b>then </b> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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"><font color="Maroon" title="c7">fI</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p2">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">fI</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c11">v</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:38_3"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <span class="p3">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">fI</font></span>)</span></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">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_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c11">v</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:38"><i><font color="Green" title="E25">A1</font></i></a>, <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="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <span class="p2">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">fI</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c11">v</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:38_3"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <span class="p3">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">fI</font></span>)</span></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">v</font>
<b>by </b><i><a class="txt" href="#E8:38_3"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">fI</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">v</font>
<b>by </b><i><a class="ref" href="vectsp11.html#T20" title="VECTSP11:th.20">VECTSP11:20</a></i>
;<br/>
<b>hence </b><a NAME="E11:38_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:38_3"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="#E5:38_3"><i><font color="Green" title="E29">A8</font></i></a>, <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="E9:38"/>
<span class="p1">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">fI</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <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"> verum</span><br/>


</div>
