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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <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>assume </b><a NAME="E1:53"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>thus </b><a NAME="E2:53"/>
<font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>1</sub></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="53_1_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:53_1_1_1_1"/><b>then </b>
 <a href="mod_2.html#K10" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="mod_2.html#D15" target="_self">Def15</a></i>;<br/><b>hence </b><a NAME="E3:53_1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:53"><i><font color="Green">E41</font></i></a>, <a class="txt" href="mod_2.html#E1:53_1_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:53_1_1_1_2"/><b>then </b>
 <a href="mod_2.html#K10" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="ref" href="mod_2.html#D15" target="_self">Def15</a></i>;<br/><b>hence </b><a NAME="E3:53_1_1_1_2"/>
<font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:53"><i><font color="Green">E41</font></i></a>, <a class="txt" href="mod_2.html#E1:53_1_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_3"><b>suppose </b></a><a NAME="E1:53_1_1_1_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:53_1_1_1_3"/><b>then </b>
 <a href="mod_2.html#K10" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="mod_2.html#D15" target="_self">Def15</a></i>;<br/><b>hence </b><a NAME="E3:53_1_1_1_3"/>
<font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:53"><i><font color="Green">E41</font></i></a>, <a class="txt" href="mod_2.html#E1:53_1_1_1_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:53_1"/>
<font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


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

<b>thus </b><a NAME="E3:53"/>
<font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:53"><i><font color="Green">E41</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>;<br/>

<b>thus </b><a NAME="E4:53"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_2_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>3</sub></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="53_2_1_1_1"><b>suppose </b></a><a NAME="E1:53_2_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1_2"><b>suppose </b></a><a NAME="E1:53_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:53_2_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>;<br/><b>hence </b><a NAME="E3:53_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1_3"><b>suppose </b></a><a NAME="E1:53_2_1_1_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:53_2_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>;<br/><b>hence </b><a NAME="E3:53_2_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1_4"><b>suppose </b></a><a NAME="E1:53_2_1_1_4"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:53_2_1_1_4"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> = 
0
<b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_4"><i><font color="Green">E42</font></i></a>, <a class="txt" href="mod_2.html#E2:53_2_1_1_4"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1_5"><b>suppose </b></a><a NAME="E1:53_2_1_1_5"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:53_2_1_1_5"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> = 
1
<b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_5"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_5"><i><font color="Green">E42</font></i></a>, <a class="txt" href="mod_2.html#E2:53_2_1_1_5"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1_6"><b>suppose </b></a><a NAME="E1:53_2_1_1_6"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:53_2_1_1_6"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> = 
1
<b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_6"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_6"><i><font color="Green">E42</font></i></a>, <a class="txt" href="mod_2.html#E2:53_2_1_1_6"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1_7"><b>suppose </b></a><a NAME="E1:53_2_1_1_7"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:53_2_1_1_7"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> = 
2
<b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_7"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_7"><i><font color="Green">E42</font></i></a>, <a class="txt" href="mod_2.html#E2:53_2_1_1_7"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1_8"><b>suppose </b></a><a NAME="E1:53_2_1_1_8"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:53_2_1_1_8"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> = 
1
<b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_8"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_8"><i><font color="Green">E42</font></i></a>, <a class="txt" href="mod_2.html#E2:53_2_1_1_8"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1_9"><b>suppose </b></a><a NAME="E1:53_2_1_1_9"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:53_2_1_1_9"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> = 
2
<b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_9"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_9"><i><font color="Green">E42</font></i></a>, <a class="txt" href="mod_2.html#E2:53_2_1_1_9"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1_10"><b>suppose </b></a><a NAME="E1:53_2_1_1_10"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:53_2_1_1_10"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> = 
2
<b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_10"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_10"><i><font color="Green">E42</font></i></a>, <a class="txt" href="mod_2.html#E2:53_2_1_1_10"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1_11"><b>suppose </b></a><a NAME="E1:53_2_1_1_11"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:53_2_1_1_11"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> = 
0
<b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_11"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">Def16</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:53_2_1_1_11"><i><font color="Green">E42</font></i></a>, <a class="txt" href="mod_2.html#E2:53_2_1_1_11"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self">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="E2:53_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mod_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>


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


</div>
