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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be    <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="vectsp_1.html#L1" title="VECTSP_1:struct.1">VectSpStr</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">f</font> being  <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> holds <br/> ( <font color="Olive" title="b2">f</font> is  <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a>  iff  ex <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Olive" title="b2">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Olive" title="b3">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">ZeroMap</a> <font color="Olive" title="b1">V</font>,<font color="Olive" title="b1">V</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="vectsp_1.html#L1" title="VECTSP_1:struct.1">VectSpStr</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <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> holds <br/> ( <font color="Olive" title="b1">f</font> is  <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a>  iff  ex <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Olive" title="b1">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Olive" title="b2">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">ZeroMap</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c2">V</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a>  iff  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Olive" title="b1">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">ZeroMap</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c2">V</font> )</span><br/>





<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Olive" title="b1">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">ZeroMap</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c2">V</font> implies <font color="Maroon" title="c3">f</font> is  <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:29_1"/>
<font color="Maroon" title="c3">f</font> is  <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Olive" title="b1">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">ZeroMap</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c2">V</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:29_1"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
for <font color="Olive" title="b1">v</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">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> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrixj2.html#D4" target="_self" title="MATRIXJ2:def.4">Def4</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">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="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <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><br/><span class="kw">assume </span><a NAME="E1:29_1_1"/>
<font color="Maroon" title="c5">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="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c4">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <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><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">v</font> = <font color="Maroon" title="c5">x</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font>

<br/>.= 
 <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="#E3:29_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c4">n</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c4">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> <font color="Maroon" title="c2">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T17" title="FUNCOP_1:th.17">FUNCOP_1:17</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> <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> <font color="Maroon" title="c2">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="grcat_1.html#K7" title="GRCAT_1:func.7">ZeroMap</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c2">V</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#D12" title="GRCAT_1:def.12">GRCAT_1:def 12</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:29_1"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Olive" title="b1">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">ZeroMap</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c2">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">given </span><font color="Maroon" title="c4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><a NAME="E3:29"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c4">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">ZeroMap</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c2">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> is  <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c4">n</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="matrixj2.html#D4" target="_self" title="MATRIXJ2:def.4">MATRIXJ2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">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> <font color="Maroon" title="c2">V</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">v</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">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> <font color="Maroon" title="c2">V</font></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v</font> = 
<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="c2">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="c2">V</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="grcat_1.html#D12" title="GRCAT_1:def.12">GRCAT_1:def 12</a></span>
<br/>.= 
 <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="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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