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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> $1 is   <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>;<br/>
<a NAME="E1:31_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c2">V1</font>
 
<b>by </b><i><a class="ref" href="vectsp11.html#T18" target="_self" title="VECTSP11:th.18">Th18</a></i>;<br/>
<a NAME="E2:31_1_1"/><b>then </b><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
;<br/>
<a NAME="E3:31_1_1"/><i><font color="Green" title="E23">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">k</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:31_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<b>then reconsider </b><font color="Maroon" title="c6">fk</font> = <font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c5">k</font> as   <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> ;<br/>
<font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c6">fk</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="vectsp11.html#T20" target="_self" title="VECTSP11:th.20">Th20</a></i>
<br/>.= 
<font color="Maroon" title="c6">fk</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c3">f</font>
<b>by </b><i><a class="ref" href="vectsp11.html#T19" target="_self" title="VECTSP11:th.19">Th19</a></i>
;<br/>
<b>hence </b><a NAME="E4:31_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <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>
<a NAME="E4:31_1_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E2:31_1_1"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="#E3:31_1_1"><i><font color="Green" title="E23">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:31_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c4">n</font> is   <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"> verum</span><br/>


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