<?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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="parsp_1.html#K9">MPS</a> <font color="Maroon">F</font></span>)</span>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="parsp_1.html#K9">MPS</a> <font color="Maroon">F</font></span>)</span>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="parsp_1.html#K9">MPS</a> <font color="Maroon">F</font></span>)</span>;<br/><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font></span><a href="zfmisc_1.html#K3">:]</a></span>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font></span><a href="zfmisc_1.html#K3">:]</a></span>;<br/><b>let </b><font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font></span><a href="zfmisc_1.html#K3">:]</a></span>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:6"/><i><font color="Green">E17</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green">E18</font></i>: 
<span class="p1"><a href="domain_1.html#K5">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K5">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">e</font>,<font color="Maroon">g</font></span><a href="mcart_1.html#K4">]</a></span>
 ;<br/>

<a NAME="E3:6"/><i><font color="Green">E45</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K5">`1</a> </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> or <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K6">`2</a> </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> or <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K7">`3</a> </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>, , </i>;<br/>
<a NAME="E4:6"/>
( <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K5">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> or <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> or <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K7">`3</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:6"/><b>then </b><i><font color="Green">E46</font></i>: 
(  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> or  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> or  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="vectsp_1.html#T44">VECTSP_1:44</a></i>;<br/>
<a NAME="E6:6"/>
<font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1"/><i><font color="Green">E47</font></i>: 
not <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:6_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K5">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</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> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</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> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K7">`3</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</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> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E3:6_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:6"/>
( <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> )
 <b>by </b><i>, <a class="txt" href="parsp_2.html#E1"><i><font color="Green">Lemma10</font></i></a>, <a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/>


</div>
