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

<b>set </b><font color="Maroon">F</font> =  <a href="mod_2.html#K18" target="_self">Z3</a> ;<br/>
<b>reconsider </b><font color="Maroon">a</font> =  <a href="rlvect_1.html#K1">0.</a> <a href="mod_2.html#K18" target="_self">Z3</a> , <font color="Maroon">b</font> =  <a href="group_1.html#K2">1.</a> <a href="mod_2.html#K18" target="_self">Z3</a>  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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <a href="mod_2.html#K18" target="_self">Z3</a> , <font color="Maroon">y</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <a href="mod_2.html#K18" target="_self">Z3</a> , <font color="Maroon">z</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <a href="mod_2.html#K18" target="_self">Z3</a> ;<br/><b>thus </b><a NAME="E1:59_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> &amp; <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> &amp; <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> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <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="mod_2.html#K18" target="_self">Z3</a>  &amp; <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="mod_2.html#K18" target="_self">Z3</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <a href="mod_2.html#K18" target="_self">Z3</a>  implies  ex <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <a href="mod_2.html#K18" target="_self">Z3</a>  st <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="mod_2.html#K18" target="_self">Z3</a>  ) &amp;  <a href="rlvect_1.html#K1">0.</a> <a href="mod_2.html#K18" target="_self">Z3</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <a href="mod_2.html#K18" target="_self">Z3</a>  &amp; <font color="Maroon">x</font> <a href="group_1.html#K1">*</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> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><b>proof </b></a><div class="add">

<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><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/>.= 
<font color="Maroon">Y</font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">X</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<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/>

<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/>

<a NAME="E5:59_1_1"/>
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</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>hence </b><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> = 
<font color="Maroon">X</font> <a href="mod_2.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><a href="mod_2.html#K10" target="_self">-</a> <font color="Maroon">X</font></span>)</span>
<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/>
;<br/><br/>

<b>thus </b><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> = 
<font color="Maroon">X</font> <a href="mod_2.html#K12" target="_self">*</a> <font color="Maroon">Y</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">Y</font> <a href="mod_2.html#K12" target="_self">*</a> <font color="Maroon">X</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="mod_2.html#K12" target="_self">*</a> <font color="Maroon">Y</font></span>)</span> <a href="mod_2.html#K12" target="_self">*</a> <font color="Maroon">Z</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">X</font> <a href="mod_2.html#K12" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="mod_2.html#K12" target="_self">*</a> <font color="Maroon">Z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="mod_2.html#K18" target="_self">Z3</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> = 
<font color="Maroon">b</font> <a href="mod_2.html#K12" target="_self">*</a> <font color="Maroon">X</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/>

<b>thus </b><a NAME="E10:59_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <a href="mod_2.html#K18" target="_self">Z3</a>  implies  ex <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <a href="mod_2.html#K18" target="_self">Z3</a>  st <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="mod_2.html#K18" target="_self">Z3</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_1_1_8"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <a href="mod_2.html#K18" target="_self">Z3</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <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> such that </b><br/><a NAME="E3:59_1_1_8"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">X</font> <a href="mod_2.html#K12" target="_self">*</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">Y</font> as   <a href="vectsp_1.html#NM3">Scalar</a> of <a href="mod_2.html#K18" target="_self">Z3</a>  ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E5:59_1_1_8"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="mod_2.html#K18" target="_self">Z3</a> 
 <b>by </b><i><a class="ref" href="mod_2.html#T6" target="_self">Th6</a>, </i>;<br/>


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

<b>thus </b><a NAME="E11:59_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <a href="mod_2.html#K18" target="_self">Z3</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <a href="mod_2.html#K18" target="_self">Z3</a> 
 <b>by </b><i/>;<br/>

<a NAME="E12:59_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="mod_2.html#K12" target="_self">*</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="mod_2.html#K12" target="_self">*</a> <font color="Maroon">Z</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">x</font> <a href="group_1.html#K1">*</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> = 
<font color="Maroon">X</font> <a href="mod_2.html#K12" 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/>.= 
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="mod_2.html#K12" target="_self">*</a> <font color="Maroon">Y</font></span>)</span> <a href="mod_2.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="mod_2.html#K12" target="_self">*</a> <font color="Maroon">Z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="mod_2.html#T6" target="_self">Th6</a>, </i>
;<br/><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> =  <a href="mod_2.html#K18" target="_self">Z3</a>  as   <a href="vectsp_1.html#NM2">Field</a> <b>by </b><i><a class="ref" href="mod_2.html#D5" target="_self">Def5</a></i>;<br/>
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> = 
<a href="mod_2.html#K13" target="_self">add3</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
2
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E5:59"/>
 <a href="mod_2.html#K18" target="_self">Z3</a>  is  <a href="vectsp_1.html#V13">Fanoian</a> <a href="vectsp_1.html#NM2">Field</a>
 <b>by </b><i>, <a class="ref" href="vectsp_1.html#D29">VECTSP_1:def 29</a></i>;<br/>


</div>
