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

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font>, <font color="Maroon" title="c4">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>;<br/>

<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>thus </b><a NAME="E2:50"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><a href="mod_2.html#K10" title="MOD_2:func.10">-</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_1_1"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1"><b>suppose </b></a><a NAME="E1:50_1_1_1_1"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:50_1_1_1_1"/><b>then </b>
 <a href="mod_2.html#K10" title="MOD_2:func.10">-</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="mod_2.html#D15" target="_self" title="MOD_2:def.15">Def15</a></i>;<br/><b>hence </b><a NAME="E3:50_1_1_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><a href="mod_2.html#K10" title="MOD_2:func.10">-</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:50"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="mod_2.html#E1:50_1_1_1_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_2"><b>suppose </b></a><a NAME="E1:50_1_1_1_2"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:50_1_1_1_2"/><b>then </b>
 <a href="mod_2.html#K10" title="MOD_2:func.10">-</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="ref" href="mod_2.html#D15" target="_self" title="MOD_2:def.15">Def15</a></i>;<br/><b>hence </b><a NAME="E3:50_1_1_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><a href="mod_2.html#K10" title="MOD_2:func.10">-</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:50"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="mod_2.html#E1:50_1_1_1_2"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_3"><b>suppose </b></a><a NAME="E1:50_1_1_1_3"/><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:50_1_1_1_3"/><b>then </b>
 <a href="mod_2.html#K10" title="MOD_2:func.10">-</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="mod_2.html#D15" target="_self" title="MOD_2:def.15">Def15</a></i>;<br/><b>hence </b><a NAME="E3:50_1_1_1_3"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><a href="mod_2.html#K10" title="MOD_2:func.10">-</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:50"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="mod_2.html#E1:50_1_1_1_3"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:50_1"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><a href="mod_2.html#K10" title="MOD_2:func.10">-</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 ;<br/>


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

<b>thus </b><a NAME="E3:50"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:50"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>;<br/>

<b>thus </b><a NAME="E4:50"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_2_1_1"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 </span><b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1_1"><b>suppose </b></a><a NAME="E1:50_2_1_1_1"/><i><font color="Green" title="E39">A5</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> = 
<font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font>
<b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_1"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1_2"><b>suppose </b></a><a NAME="E1:50_2_1_1_2"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:50_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>;<br/><b>hence </b><a NAME="E3:50_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1_3"><b>suppose </b></a><a NAME="E1:50_2_1_1_3"/><i><font color="Green" title="E39">A6</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:50_2_1_1_3"/><b>then </b>
<font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>;<br/><b>hence </b><a NAME="E3:50_2_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_3"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1_4"><b>suppose </b></a><a NAME="E1:50_2_1_1_4"/><i><font color="Green" title="E39">A7</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:50_2_1_1_4"/><b>then </b><i><font color="Green" title="E40">A8</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_4"><i><font color="Green" title="E39">A7</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_4"><i><font color="Green" title="E39">A7</font></i></a>, <a class="txt" href="mod_2.html#E2:50_2_1_1_4"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1_5"><b>suppose </b></a><a NAME="E1:50_2_1_1_5"/><i><font color="Green" title="E39">A9</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:50_2_1_1_5"/><b>then </b><i><font color="Green" title="E40">A10</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> = 
1
<b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_5"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_5"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="mod_2.html#E2:50_2_1_1_5"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1_6"><b>suppose </b></a><a NAME="E1:50_2_1_1_6"/><i><font color="Green" title="E39">A11</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:50_2_1_1_6"/><b>then </b><i><font color="Green" title="E40">A12</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> = 
1
<b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_6"><i><font color="Green" title="E39">A11</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_6"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="mod_2.html#E2:50_2_1_1_6"><i><font color="Green" title="E40">A12</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1_7"><b>suppose </b></a><a NAME="E1:50_2_1_1_7"/><i><font color="Green" title="E39">A13</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:50_2_1_1_7"/><b>then </b><i><font color="Green" title="E40">A14</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> = 
2
<b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_7"><i><font color="Green" title="E39">A13</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_7"><i><font color="Green" title="E39">A13</font></i></a>, <a class="txt" href="mod_2.html#E2:50_2_1_1_7"><i><font color="Green" title="E40">A14</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1_8"><b>suppose </b></a><a NAME="E1:50_2_1_1_8"/><i><font color="Green" title="E39">A15</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:50_2_1_1_8"/><b>then </b><i><font color="Green" title="E40">A16</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> = 
1
<b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_8"><i><font color="Green" title="E39">A15</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_8"><i><font color="Green" title="E39">A15</font></i></a>, <a class="txt" href="mod_2.html#E2:50_2_1_1_8"><i><font color="Green" title="E40">A16</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1_9"><b>suppose </b></a><a NAME="E1:50_2_1_1_9"/><i><font color="Green" title="E39">A17</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:50_2_1_1_9"/><b>then </b><i><font color="Green" title="E40">A18</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> = 
2
<b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_9"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_9"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="mod_2.html#E2:50_2_1_1_9"><i><font color="Green" title="E40">A18</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1_10"><b>suppose </b></a><a NAME="E1:50_2_1_1_10"/><i><font color="Green" title="E39">A19</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:50_2_1_1_10"/><b>then </b><i><font color="Green" title="E40">A20</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> = 
2
<b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_10"><i><font color="Green" title="E39">A19</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_10"><i><font color="Green" title="E39">A19</font></i></a>, <a class="txt" href="mod_2.html#E2:50_2_1_1_10"><i><font color="Green" title="E40">A20</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1_11"><b>suppose </b></a><a NAME="E1:50_2_1_1_11"/><i><font color="Green" title="E39">A21</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:50_2_1_1_11"/><b>then </b><i><font color="Green" title="E40">A22</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_11"><i><font color="Green" title="E39">A21</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:50_2_1_1_11"><i><font color="Green" title="E39">A21</font></i></a>, <a class="txt" href="mod_2.html#E2:50_2_1_1_11"><i><font color="Green" title="E40">A22</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:50_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span>
 ;<br/>


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


</div>
