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<b>let </b><font color="Maroon">F</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">S</font> be   <a href="symsp_1.html#NM1">SymSp</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>





<b>assume </b><a NAME="E1:25"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
 ;<br/>

<b>set </b><font color="Maroon">1F</font> =  <a href="group_1.html#K2">1.</a> <font color="Maroon">F</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 ;<br/><b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:25_1"/><i><font color="Green">E31</font></i>: 
not <font color="Maroon">c</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">k</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E5:25_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c</font></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:25_1"/>
<font color="Maroon">a</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="symsp_1.html#E5"><i><font color="Green">Lemma23</font></i></a>, </i>;<br/><a NAME="E7:25_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> &amp; <font color="Maroon">a</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>;<br/><a NAME="E8:25_1"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c</font></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><a NAME="E9:25_1"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c</font></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/><a NAME="E10:25_1"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c</font></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/><a NAME="E11:25_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c</font></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T59">VECTSP_1:59</a></i>;<br/><a NAME="E12:25_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c</font></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/><a NAME="E13:25_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">k</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/><a NAME="E14:25_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T41">VECTSP_1:41</a></i>;<br/><a NAME="E15:25_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>;<br/><a NAME="E16:25_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/><a NAME="E17:25_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/><a NAME="E18:25_1"/><b>then </b>
<font color="Maroon">a</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="vectsp_1.html#T67">VECTSP_1:67</a></i>;<br/><a NAME="E19:25_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><a NAME="E20:25_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c</font></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E5"><i><font color="Green">Lemma23</font></i></a>, <a class="ref" href="vectsp_1.html#T68">VECTSP_1:68</a></i>;<br/><b>hence </b><a NAME="E21:25_1"/>
<font color="Maroon">a</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:25"/>
<font color="Maroon">a</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


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