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<b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:6_1"/><i><font color="Green">E38</font></i>: 
not <font color="Maroon">i</font> is <a href="abian.html#V1" target="_self">even</a>
 ;<br/><b>assume </b><a NAME="E2:6_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">j</font> being  <a href="int_1.html#NM1">Integer</a>  holds <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:6_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="int_1.html#T8">INT_1:8</a></i>;<br/><b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:6_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="scheme1.html#T1">SCHEME1:1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> )
 ;<br/><div class="add"><a NAME="E2:6_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E3:6_1_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_3"><b>suppose </b></a><a NAME="E1:6_1_1_3"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_1_1_3"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_4"><b>suppose </b></a><a NAME="E1:6_1_1_4"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:6_1_1_4"/><b>then </b>
<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> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><b>hence </b><a NAME="E3:6_1_1_4"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">j</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><a NAME="E2:6"/><i><font color="Green">E43</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">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<b>given </b><font color="Maroon">k</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><a NAME="E3:6"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="abian.html#D1" target="_self">ABIAN:def 1</a><br/>

<a NAME="E4:6"/>
1 <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#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:6"/><b>then </b>
1 <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E6:6"/>
contradiction
 <b>by </b><i><a class="ref" href="int_1.html#T22">INT_1:22</a></i>;<br/>


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