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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="ortsp_1.html#NM1">OrtSp</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</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>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">k</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">l</font> be   <a href="struct_0.html#NM1">Element</a> of <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/>
<b>assume </b><a NAME="E1:15"/><i><font color="Green">E22</font></i>: 
( not <font color="Maroon">b</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x</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">b</font></span>)</span> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font></span>)</span> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">a</font> )
 ;<br/>

<b>assume </b><a NAME="E2:15"/><i><font color="Green">E26</font></i>: 
not <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font>
 ;<br/>

<a NAME="E3:15"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font></span>)</span> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="txt" href="ortsp_1.html#E4"><i><font color="Green">Lemma13</font></i></a></i>;<br/>
<a NAME="E4:15"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font></span>)</span> <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">l</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="symsp_1.html#R1">_|_</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="E5:15"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font></span>)</span> <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> <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> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#T59">VECTSP_1:59</a></i>;<br/>
<a NAME="E6:15"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<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">l</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font></span>)</span> <a href="symsp_1.html#R1">_|_</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="E7:15"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font></span>)</span> <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> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="group_1.html#K10">*</a> <span class="p4">(<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">b</font></span>)</span> <a href="symsp_1.html#R1">_|_</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="E8:15"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font></span>)</span> <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">l</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font></span>)</span> <a href="symsp_1.html#R1">_|_</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="E9:15"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font> <a href="symsp_1.html#R1">_|_</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="E10:15"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">l</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"><font color="Maroon">k</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font></span>)</span> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:15"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font> <a href="symsp_1.html#R1">_|_</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="E12:15"/>
<font color="Maroon">k</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="ortsp_1.html#E5"><i><font color="Green">Lemma16</font></i></a>, <a class="ref" href="rlvect_1.html#T35">RLVECT_1:35</a></i>;<br/>
<a NAME="E13:15"/><b>then </b>
<font color="Maroon">k</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">l</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>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>;<br/>
<a NAME="E14:15"/><b>then </b>
<span class="p1">(<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">b</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>;<br/>
<b>hence </b><a NAME="E15:15"/>
contradiction
 <b>by </b><i>, <a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/>


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