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

<b>set </b><font color="Maroon">A</font> = <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2;<br/>
<b>set </b><font color="Maroon">M</font> = <font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> 2;<br/>
<a NAME="E1:8"/>
<font color="Maroon">i</font> <a href="int_1.html#K5">div</a> 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
 
<b>by </b><i><a class="ref" href="int_1.html#D7">INT_1:def 7</a></i>;<br/>
<a NAME="E2:8"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="int_1.html#K5">div</a> 2 &amp; <font color="Maroon">i</font> <a href="int_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K7">/</a> 2 )
 
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E3:8"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E4:8"/><b>then </b>
(  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span> &amp;  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E5:8"/><b>then </b>
( <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span></span>)</span> &amp; <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:8"/><b>then </b>
( 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2</span>)</span> )
 
;<br/>
<a NAME="E7:8"/><b>then </b><i><font color="Green">E31</font></i>: 
( 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> 2 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> 2 )
 
<b>by </b><i><a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> 2 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E9:8"/>
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E10:8"/><b>then </b>
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a> 2
 
<b>by </b><i><a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>;<br/>
<b>hence </b><a NAME="E11:8"/>
( <font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


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