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

<b>let </b><font color="Maroon" title="c4">F</font> be   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c5">S</font> be   <a href="symsp_1.html#NM1" title="SYMSP_1:NM.1">SymSp</a> of <font color="Maroon" title="c4">F</font>;<br/><b>let </b><font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font>;<br/>











<b>set </b><font color="Maroon" title="c10">0F</font> =  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green" title="E43">A1</font></i>: 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">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="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font>
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green" title="E44">A2</font></i>: 
not <font color="Maroon" title="c6">b</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c7">a</font>
 ;<br/>

<a NAME="E3:48"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c5">S</font>
 
<b>by </b><i><a class="txt" href="symsp_1.html#E2:48"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="symsp_1.html#E62"><i><font color="Green" title="E41">Lm6</font></i></a></i>;<br/>
<a NAME="E4:48"/><i><font color="Green" title="E46">A4</font></i>: 
(  <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font> implies <font color="Maroon" title="c9">y</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1"/><i><font color="Green" title="E47">A5</font></i>: 
 <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font>
 ;<br/>

<div><i><font color="Green" title="E48">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_1"/>
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font>  holds <br/>( <font color="Olive" title="b1">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c7">a</font> or <font color="Olive" title="b1">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">x</font> )
 ;<br/><a NAME="E2:48_1_1"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c5">S</font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E3:48"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="symsp_1.html#T21" target="_self" title="SYMSP_1:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E3:48_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E62"><i><font color="Green" title="E41">Lm6</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c11">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font><b> such that </b><a NAME="E1:48_1_2"/><i><font color="Green" title="E49">A7</font></i>: 
( not <font color="Maroon" title="c11">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c7">a</font> &amp; not <font color="Maroon" title="c11">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">x</font> )
 ;<br/><a NAME="E2:48_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c11">p</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c9">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:48"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="symsp_1.html#E2:48"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="symsp_1.html#E1:48_1"><i><font color="Green" title="E47">A5</font></i></a>, <a class="txt" href="symsp_1.html#E1:48_1_2"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="symsp_1.html#D7" target="_self" title="SYMSP_1:def.7">Def7</a></i>;<br/><a NAME="E3:48_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c11">p</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font> or  <a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font> )
 <b>by </b><i><a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>;<br/><a NAME="E4:48_1_2"/><b>then </b><i><font color="Green" title="E50">A8</font></i>: 
(  <a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font> or  <a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font> or  <a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font> )
 <b>by </b><i><a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_2_1"/><i><font color="Green" title="E51">A9</font></i>: 
 <a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font>
 ;<br/><a NAME="E2:48_1_2_1"/>
not <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c11">p</font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:48_1_2"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="symsp_1.html#T12" target="_self" title="SYMSP_1:th.12">Th12</a></i>;<br/><a NAME="E3:48_1_2_1"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c11">p</font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:48_1_2_1"><i><font color="Green" title="E51">A9</font></i></a>, <a class="ref" href="symsp_1.html#T36" target="_self" title="SYMSP_1:th.36">Th36</a></i>;<br/><b>hence </b><a NAME="E4:48_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:48_1_2"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="symsp_1.html#T12" target="_self" title="SYMSP_1:th.12">Th12</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:48_1_2"/>
<font color="Maroon" title="c9">y</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E2:48"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="symsp_1.html#E1:48_1_2"><i><font color="Green" title="E49">A7</font></i></a>, <a class="txt" href="symsp_1.html#E4:48_1_2"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="symsp_1.html#T36" target="_self" title="SYMSP_1:th.36">Th36</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:48_1"/>
<font color="Maroon" title="c9">y</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E2:48_1"><i><font color="Green" title="E48">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:48"/>
( <font color="Maroon" title="c9">y</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">x</font> implies  <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_2"/><i><font color="Green" title="E47">A10</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">x</font>
 ;<br/>

<div><i><font color="Green" title="E48">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c5">S</font>
 ;<br/><a NAME="E2:48_2_1"/><b>then </b>
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font>  holds <br/>( <font color="Olive" title="b1">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c7">a</font> or <font color="Olive" title="b1">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="txt" href="symsp_1.html#E62"><i><font color="Green" title="E41">Lm6</font></i></a></i>;<br/><b>hence </b><a NAME="E3:48_2_1"/>
 <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:48"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="symsp_1.html#E2:48"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="symsp_1.html#D7" target="_self" title="SYMSP_1:def.7">Def7</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_2_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c5">S</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c11">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font><b> such that </b><br/><a NAME="E3:48_2_2"/><i><font color="Green" title="E49">A12</font></i>: 
( not <font color="Maroon" title="c11">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c7">a</font> &amp; not <font color="Maroon" title="c11">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="txt" href="symsp_1.html#E3:48"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="symsp_1.html#T21" target="_self" title="SYMSP_1:th.21">Th21</a></i>;<br/><a NAME="E4:48_2_2"/>
 <a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:48_2"><i><font color="Green" title="E47">A10</font></i></a>, <a class="txt" href="symsp_1.html#E3:48_2_2"><i><font color="Green" title="E49">A12</font></i></a>, <a class="ref" href="symsp_1.html#T36" target="_self" title="SYMSP_1:th.36">Th36</a></i>;<br/><a NAME="E5:48_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c11">p</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c9">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>;<br/><b>hence </b><a NAME="E6:48_2_2"/>
 <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:48"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="symsp_1.html#E2:48"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="symsp_1.html#E3:48_2_2"><i><font color="Green" title="E49">A12</font></i></a>, <a class="ref" href="symsp_1.html#D7" target="_self" title="SYMSP_1:def.7">Def7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:48_2"/>
 <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E2:48_2"><i><font color="Green" title="E48">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:48"/>
(  <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font> iff <font color="Maroon" title="c9">y</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="txt" href="symsp_1.html#E4:48"><i><font color="Green" title="E46">A4</font></i></a></i>;<br/>


</div>
