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

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:32"/>
( not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> )
 ;<br/>

<a NAME="E2:32"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">i</font> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_2.html#T24">NAT_2:24</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:32"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 )
 <b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:32"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 )
 <b>by </b><i>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<a NAME="E7:32"/>
<font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:32"/>
not <font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font> is <a href="abian.html#V1">even</a>
 ;<br/>


</div>
