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

<b>thus </b><a NAME="E1:54_1_1"/>
 <a href="mod_2.html#K18" target="_self">Z3</a>  is <a href="rlvect_1.html#V4">add-associative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_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="mod_2.html#K18" target="_self">Z3</a> , <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="mod_2.html#K18" target="_self">Z3</a> , <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="mod_2.html#K18" target="_self">Z3</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a><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="enumset1.html#K1">{</a><span class="default">0,1,2</span><a href="enumset1.html#K1">}</a></span> ;<br/>
<b>thus </b><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> = 
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">Y</font></span>)</span> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">Z</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">X</font> <a href="mod_2.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">Z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<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/>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:54_1_1"/>
 <a href="mod_2.html#K18" target="_self">Z3</a>  is <a href="rlvect_1.html#V5">right_zeroed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_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="mod_2.html#K18" target="_self">Z3</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a><br/>

<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">x</font>, <font color="Maroon">a</font> = 0 as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="enumset1.html#K1">{</a><span class="default">0,1,2</span><a href="enumset1.html#K1">}</a></span> <b>by </b><i/>;<br/>
<b>thus </b><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="mod_2.html#K18" target="_self">Z3</a> </span>)</span> = 
<font color="Maroon">X</font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">a</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="mod_2.html#K18" target="_self">Z3</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><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="enumset1.html#K1">{</a><span class="default">0,1,2</span><a href="enumset1.html#K1">}</a></span> ;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <a href="mod_2.html#K15" target="_self">compl3</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="mod_2.html#K18" target="_self">Z3</a>  ;<br/>
<b>reconsider </b><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="enumset1.html#K1">{</a><span class="default">0,1,2</span><a href="enumset1.html#K1">}</a></span> ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<a NAME="E6:54_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a>  <a href="mod_2.html#K10" target="_self">-</a> <font color="Maroon">x'</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E7:54_1_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="mod_2.html#K18" target="_self">Z3</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_1_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:54_1_1_3_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:54_1_1_3_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, </i>;<br/><b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> = 
<font color="Maroon">x'</font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">y'</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="mod_2.html#K18" target="_self">Z3</a> 
<b>by </b><i>, , <a class="txt" href="mod_2.html#E6:54_1_1"><i><font color="Green">E29</font></i></a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:54_1_1_3_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:54_1_1_3_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i>, </i>;<br/><b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> = 
<font color="Maroon">x'</font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">y'</font>
<b>by </b><i><a class="txt" href="mod_2.html#E2:54_1_1_3_1_2"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="mod_2.html#K18" target="_self">Z3</a> 
<b>by </b><i>, , <a class="txt" href="mod_2.html#E6:54_1_1"><i><font color="Green">E29</font></i></a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_3_1_3"><b>suppose </b></a><a NAME="E1:54_1_1_3_1_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:54_1_1_3_1_3"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, </i>;<br/><b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> = 
<font color="Maroon">x'</font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">y'</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="mod_2.html#K18" target="_self">Z3</a> 
<b>by </b><i>, , <a class="txt" href="mod_2.html#E6:54_1_1"><i><font color="Green">E29</font></i></a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div>
