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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">t</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:24"/><i><font color="Green">E25</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font>
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E3:24"/><i><font color="Green">E29</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 ;<br/>

<b>thus </b><a NAME="E4:24"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">t</font> is <a href="abian.html#V1">even</a> implies <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K5">div</a> <font color="Maroon">t</font> is <a href="abian.html#V1">even</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">t</font> is <a href="abian.html#V1">even</a>
 ;<br/>

<b>then consider </b><font color="Maroon">p</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="E3:24_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="abian.html#D2">ABIAN:def 2</a></i>;<br/>
<a NAME="E4:24_1"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E5:24_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<b>then consider </b><font color="Maroon">q</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:24_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E8:24_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">q</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D1">NAT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:24_1"/><i><font color="Green">E34</font></i>: 
<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">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E11:24_1"/><i><font color="Green">E45</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E12:24_1"/><b>then </b><i><font color="Green">E49</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E13:24_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , , <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E14:24_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E15:24_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E16:24_1"/>
1 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E17:24_1"/><b>then </b>
<font color="Maroon">t</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E18:24_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">q</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:24_1_1_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>hence </b><a NAME="E3:24_1_1_1_1"/>
<font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i>, <a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_1_1_2"/>
<font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E20:24_1"/><b>then </b>
<font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E21:24_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E22:24_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E23:24_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">q</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">q</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">q</font>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<b>hence </b><a NAME="E25:24_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K5">div</a> <font color="Maroon">t</font> is <a href="abian.html#V1">even</a>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D1">NAT_1:def 1</a></i>;<br/>


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

<b>assume </b><a NAME="E5:24"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K5">div</a> <font color="Maroon">t</font> is <a href="abian.html#V1">even</a>
 ;<br/>

<b>then consider </b><font color="Maroon">p</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="E7:24"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K5">div</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="abian.html#D2">ABIAN:def 2</a></i>;<br/>
<a NAME="E8:24"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E9:24"/><b>then </b>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<b>then consider </b><font color="Maroon">q</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:24"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E12:24"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">q</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D1">NAT_1:def 1</a></i>;<br/>
<a NAME="E13:24"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">q</font>
 
<b>by </b><i>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E15:24"/><i><font color="Green">E57</font></i>: 
<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">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E16:24"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">q</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E17:24"/>
<font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">t</font> is <a href="abian.html#V1">even</a>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D1">NAT_1:def 1</a></i>;<br/>


</div>
