<?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">mF</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>;<br/><b>let </b><font color="Maroon">mo</font> be   <a href="relset_1.html#NM3">Relation</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>;<br/>





<b>set </b><font color="Maroon">H</font> =  <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #);<br/>
<a NAME="E1:6"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #) holds  <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #), <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #);<br/>

<a NAME="E1:6_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">md</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> ;<br/>
<a NAME="E3:6_1"/>
( <font color="Maroon">md</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp; <font color="Maroon">md</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:6_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #) holds  <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">z</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #), <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #), <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #);<br/>

<a NAME="E1:6_2"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">md</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>;<br/>
<a NAME="E2:6_2"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">md</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>;<br/>
<a NAME="E3:6_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">md</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">md</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font>, <font color="Maroon">z</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> ;<br/>
<a NAME="E5:6_2"/>
<font color="Maroon">md</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">md</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:6_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #) holds  <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #)</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #);<br/>

<a NAME="E1:6_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #)</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">md</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #)</span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> ;<br/>
<a NAME="E3:6_3"/>
<font color="Maroon">md</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #)</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:6_3"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #)</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:6"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #)  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #) st <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #);<br/>

<a NAME="E1:6_4"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">md</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>;<br/>
<a NAME="E2:6_4"/><i><font color="Green">E39</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>;<br/>
<b>take </b>
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E3:6_4"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #)
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:6"/>
(  <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #) is <a href="rlvect_1.html#V3">Abelian</a> &amp;  <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #) is <a href="rlvect_1.html#V4">add-associative</a> &amp;  <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #) is <a href="rlvect_1.html#V5">right_zeroed</a> &amp;  <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">md</font>,<font color="Maroon">o</font>,<font color="Maroon">mF</font>,<font color="Maroon">mo</font> #) is <a href="rlvect_1.html#V6">right_complementable</a> )
 <b>by </b><i>, , , <a class="ref" href="vectsp_1.html#T7">VECTSP_1:7</a></i>;<br/>


</div>
