<?xml version="1.0"?>
<div class="add">

<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>set </b><font color="Maroon">1F</font> =  <a href="group_1.html#K2">1.</a> <font color="Maroon">F</font>;<br/>
<b>set </b><font color="Maroon">0V</font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:17"/><i><font color="Green">E22</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>
 <b>and </b><br/><a NAME="E2:17"/><i><font color="Green">E26</font></i>: 
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <font color="Olive">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>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:17"/><i><font color="Green">E27</font></i>: 
<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>
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E5:17"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <font color="Olive">b</font> <a href="symsp_1.html#R1">_|_</a> <font color="Olive">b</font>
 ;<br/>

<a NAME="E6:17"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <font color="Olive">c</font> <a href="symsp_1.html#R1">_|_</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>



<a NAME="E1:17_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">d</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font> &amp; <font color="Maroon">c</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">c</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:17_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:17_1"/>
<font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:17_1"/><b>then </b>
<font color="Maroon">d</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:17_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="txt" href="ortsp_1.html#E1:17"><i><font color="Green">E22</font></i></a>, </i>;<br/>
<a NAME="E6:17_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</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">c</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">d</font></span>)</span> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>;<br/>
<a NAME="E7:17_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">d</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E8:17_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">d</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E9:17_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">d</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>;<br/>
<a NAME="E10:17_1"/><b>then </b>
<font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">d</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/>
<a NAME="E11:17_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="vectsp_1.html#K6">*</a> <font color="Maroon">d</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">d</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/>
<a NAME="E12:17_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="vectsp_1.html#K6">*</a> <font color="Maroon">d</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">d</font></span>)</span> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/>
<a NAME="E13:17_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">d</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/>
<a NAME="E14:17_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">d</font></span>)</span> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:17_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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="p4">(<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="group_1.html#K10">*</a> <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></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">d</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/>
<a NAME="E16:17_1"/><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">d</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</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="vectsp_1.html#D22">VECTSP_1:def 22</a></i>;<br/>
<a NAME="E17:17_1"/><b>then </b>
<font color="Maroon">d</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/>
<a NAME="E18:17_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font>
 
<b>by </b><i/>;<br/>
<a NAME="E19:17_1"/>
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">d</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E20:17_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font></span>)</span> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="txt" href="ortsp_1.html#E6"><i><font color="Green">Lemma23</font></i></a>, </i>;<br/>
<a NAME="E21:17_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">d</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E22:17_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>;<br/>
<b>hence </b><a NAME="E23:17_1"/>
<font color="Maroon">c</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:17"/>
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <br/>( not <font color="Olive">c</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">a</font> &amp; not <font color="Olive">c</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">a</font> &amp; not <font color="Olive">c</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">a</font> &amp; not <font color="Olive">c</font> <a href="symsp_1.html#R1">_|_</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:17"/>
contradiction
 <b>by </b><i/>;<br/>


</div>
