<?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> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><b>suppose </b></a><a NAME="E1:19_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>thus </b><a NAME="E3:19_1_1"/>
( ( <font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a> iff <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a> ) iff <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:19_1_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:19_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font> is <a href="abian.html#V1">even</a>
 ;<br/><b>hence </b><a NAME="E4:19_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a> iff <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a> ) iff <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_1_1_1_2"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:19_1_1_1_1_2"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font> is <a href="abian.html#V1">even</a>
 ;<br/><b>hence </b><a NAME="E4:19_1_1_1_1_2"/>
( ( <font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a> iff <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a> ) iff <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2"><b>suppose </b></a><a NAME="E1:19_1_2"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>thus </b><a NAME="E3:19_1_2"/>
( ( <font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a> iff <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a> ) iff <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_2_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:19_1_2_1_1_1"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font> is <a href="abian.html#V1">even</a>
 ;<br/><b>hence </b><a NAME="E4:19_1_2_1_1_1"/>
( ( <font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a> iff <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a> ) iff <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_2_1_1_2"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:19_1_2_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font> is <a href="abian.html#V1">even</a>
 ;<br/><b>hence </b><a NAME="E4:19_1_2_1_1_2"/>
( ( <font color="Maroon">c<sub>1</sub></font> is <a href="abian.html#V1">even</a> iff <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a> ) iff <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> is <a href="abian.html#V1">even</a> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div>
