<?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">d</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>let </b><font color="Maroon">h</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="E1:12"/><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="E2:12"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">b</font>,<font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E4:12"/><i><font color="Green">E46</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">c</font>,<font color="Maroon">d</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">g</font>,<font color="Maroon">h</font></span><a href="mcart_1.html#K4">]</a></span>
 ;<br/>

<a NAME="E5:12"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T33">MCART_1:33</a></i>;<br/>
<a NAME="E6:12"/><b>then </b><i><font color="Green">E48</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="E7:12"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K6">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K7">`3</a> </span>)</span></span><a href="domain_1.html#K4">]</a></span> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K6">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K7">`3</a> </span>)</span></span><a href="domain_1.html#K4">]</a></span> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K6">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K7">`3</a> </span>)</span></span><a href="domain_1.html#K4">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T48">MCART_1:48</a></i>;<br/>
<a NAME="E8:12"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">e</font> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="mcart_1.html#K5">`1</a>  or <font color="Maroon">e</font> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="mcart_1.html#K6">`2</a>  or <font color="Maroon">e</font> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="mcart_1.html#K7">`3</a>  )
 
<b>by </b><i>, <a class="txt" href="parsp_2.html#E2:12"><i><font color="Green">E18</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <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>, <font color="Maroon">j</font> being    <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>, <font color="Maroon">k</font> being    <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>, <font color="Maroon">l</font> being    <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><b> such that </b><br/><a NAME="E10:12"/><i><font color="Green">E51</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">c</font>,<font color="Maroon">d</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">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font>,<font color="Maroon">l</font></span><a href="mcart_1.html#K4">]</a></span> &amp; (  ex <font color="Olive">K</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font> st <br/>( <font color="Olive">K</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</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">j</font> <a href="mcart_1.html#K5">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</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">l</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> &amp; <font color="Olive">K</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</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">j</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</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">l</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> &amp; <font color="Olive">K</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</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">j</font> <a href="mcart_1.html#K7">`3</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</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">l</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> ) or ( <span class="p1">(<span class="default"><font color="Maroon">i</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">j</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</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">j</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</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">j</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> ) ) )
 <b>by </b><i><a class="txt" href="parsp_2.html#E2"><i><font color="Green">Lemma14</font></i></a>, </i>;<br/>
<a NAME="E11:12"/>
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> )
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T33">MCART_1:33</a></i>;<br/>
<b>then consider </b><font color="Maroon">K</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E13:12"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">K</font> <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#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</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">h</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> &amp; <font color="Maroon">K</font> <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#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</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">h</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> &amp; <font color="Maroon">K</font> <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#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</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">h</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E14:12"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">e</font> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="mcart_1.html#K5">`1</a>  or <font color="Maroon">e</font> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="mcart_1.html#K6">`2</a>  or <font color="Maroon">e</font> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="mcart_1.html#K7">`3</a>  )
 
<b>by </b><i>, <a class="txt" href="parsp_2.html#E2:12"><i><font color="Green">E18</font></i></a>, <a class="ref" href="parsp_2.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>consider </b><font color="Maroon">m</font> being    <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>, <font color="Maroon">n</font> being    <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>, <font color="Maroon">o</font> being    <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>, <font color="Maroon">w</font> being    <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><b> such that </b><br/><a NAME="E16:12"/><i><font color="Green">E54</font></i>: 
( <span class="p1"><a href="domain_1.html#K5">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">d</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">m</font>,<font color="Maroon">n</font>,<font color="Maroon">o</font>,<font color="Maroon">w</font></span><a href="mcart_1.html#K4">]</a></span> &amp; (  ex <font color="Olive">L</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font> st <br/>( <font color="Olive">L</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</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">n</font> <a href="mcart_1.html#K5">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o</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">w</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> &amp; <font color="Olive">L</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</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">n</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o</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">w</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> &amp; <font color="Olive">L</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</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">n</font> <a href="mcart_1.html#K7">`3</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o</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">w</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> ) or ( <span class="p1">(<span class="default"><font color="Maroon">m</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">n</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">m</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">n</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">m</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">n</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> ) ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E17:12"/>
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> )
 
<b>by </b><i><a class="txt" href="parsp_2.html#E1:12"><i><font color="Green">E17</font></i></a>, , <a class="ref" href="mcart_1.html#T33">MCART_1:33</a></i>;<br/>
<b>then consider </b><font color="Maroon">L</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E19:12"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">L</font> <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">g</font> <a href="mcart_1.html#K5">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</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">h</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> &amp; <font color="Maroon">L</font> <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">g</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</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">h</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> &amp; <font color="Maroon">L</font> <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">g</font> <a href="mcart_1.html#K7">`3</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</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">h</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E20:12"/>
( <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K5">`1</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K5">`1</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</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> &amp; <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</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> &amp; <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K7">`3</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K7">`3</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</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> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E21:12"/><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> <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="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#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rlvect_1.html#K4">+</a> <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></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">g</font> <a href="mcart_1.html#K5">`1</a> </span>)</span></span>)</span> &amp; <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> <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="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#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rlvect_1.html#K4">+</a> <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></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">g</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> &amp; <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> <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="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#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rlvect_1.html#K4">+</a> <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></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">g</font> <a href="mcart_1.html#K7">`3</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="parsp_2.html#E1"><i><font color="Green">Lemma10</font></i></a></i>;<br/>
<a NAME="E22:12"/><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> <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="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> &amp; <font color="Maroon">L</font> <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"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="txt" href="parsp_2.html#E2:12"><i><font color="Green">E18</font></i></a>, </i>;<br/>
<a NAME="E23:12"/><b>then </b>
( <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#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</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">h</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> &amp; <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#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</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">h</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> &amp; <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#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</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">h</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E24:12"/><b>then </b>
( <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#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</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">h</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> &amp; <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#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</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">h</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> &amp; <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#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</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">h</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>;<br/>
<b>hence </b><a NAME="E25:12"/>
( <font color="Maroon">h</font> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K5">`1</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> &amp; <font color="Maroon">h</font> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> &amp; <font color="Maroon">h</font> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K7">`3</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> )
 <b>by </b><i/>;<br/>


</div>
