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

<span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</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">S</font> being  <a href="ortsp_1.html#NM1" title="ORTSP_1:NM.1">OrtSp</a> of <font color="Maroon" title="c4">F</font>  st <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> &amp;  ex <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">S</font> st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">S</font> holds <br/> ex <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">S</font> st not <font color="Olive" title="b2">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">S</font> be   <a href="ortsp_1.html#NM1" title="ORTSP_1:NM.1">OrtSp</a> of <font color="Maroon" title="c4">F</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"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> &amp;  ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</font> implies  ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st not <font color="Olive" title="b1">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c6">1F</font> =  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">F</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">0V</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:29"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">and </span><br/><a NAME="E2:29"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> holds  <font color="Olive" title="b1">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font><span class="kw"> such that </span><br/><a NAME="E4:29"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:29"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> holds  <font color="Olive" title="b1">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:29"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> holds  <font color="Olive" title="b2">d</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c10">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</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="c10">d</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> </span><br/>



<a NAME="E1:29_1"/>
( <font color="Maroon" title="c10">d</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; <font color="Maroon" title="c9">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>;<br/>
<a NAME="E2:29_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">d</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ortsp_1.html#T21" target="_self" title="ORTSP_1:th.21">Th21</a></span>;<br/>
<a NAME="E3:29_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c10">d</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ortsp_1.html#T12" target="_self" title="ORTSP_1:th.12">Th12</a></span>;<br/>
<a NAME="E4:29_1"/>
<font color="Maroon" title="c10">d</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>;<br/>
<a NAME="E5:29_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">d</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="ortsp_1.html#D2" target="_self" title="ORTSP_1:def.2">Def2</a></span>;<br/>
<a NAME="E6:29_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">d</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>;<br/>
<a NAME="E7:29_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">d</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>;<br/>
<a NAME="E8:29_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">d</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>;<br/>
<a NAME="E9:29_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">d</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>;<br/>
<a NAME="E10:29_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">d</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D29" title="VECTSP_1:def.29">VECTSP_1:def 29</a></span>;<br/>
<a NAME="E11:29_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">d</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D29" title="VECTSP_1:def.29">VECTSP_1:def 29</a></span>;<br/>
<a NAME="E12:29_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">d</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D27" title="VECTSP_1:def.27">VECTSP_1:def 27</a></span>;<br/>
<a NAME="E13:29_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">d</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ortsp_1.html#D2" target="_self" title="ORTSP_1:def.2">Def2</a></span>;<br/>
<a NAME="E14:29_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">d</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D28" title="VECTSP_1:def.28">VECTSP_1:def 28</a>, <a class="ref" href="vectsp_1.html#D29" title="VECTSP_1:def.29">VECTSP_1:def 29</a></span>;<br/>
<a NAME="E15:29_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">d</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="vectsp_1.html#D22" title="VECTSP_1:def.22">VECTSP_1:def 22</a></span>;<br/>
<a NAME="E16:29_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">d</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D29" title="VECTSP_1:def.29">VECTSP_1:def 29</a></span>;<br/>
<a NAME="E17:29_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c10">d</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ortsp_1.html#T12" target="_self" title="ORTSP_1:th.12">Th12</a></span>;<br/>
<a NAME="E18:29_1"/>
<font color="Maroon" title="c10">d</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="ortsp_1.html#T16" target="_self" title="ORTSP_1:th.16">Th16</a></span>;<br/>
<a NAME="E19:29_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">d</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:29_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="ortsp_1.html#D2" target="_self" title="ORTSP_1:def.2">Def2</a></span>;<br/>
<a NAME="E20:29_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">d</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>;<br/>
<a NAME="E21:29_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">d</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:29_1"/>
<font color="Maroon" title="c10">d</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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>
<a NAME="E7:29"/>
 ex <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st <br/>( not <font color="Maroon" title="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="ortsp_1.html#D2" target="_self" title="ORTSP_1:def.2">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:29"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29"><span class="lab"><font color="Green" title="E20">A5</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>
