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

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><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> ;<br/><b>let </b><font color="Maroon">D</font> be  <a href="compts_1.html#V6">compact</a> non <a href="sppol_1.html#V1">horizontal</a> non <a href="sppol_1.html#V2">vertical</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">k</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="E1:27"/><i><font color="Green">E48</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span>
 <b>and </b><br/><a NAME="E3:27"/><i><font color="Green">E50</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:27"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>,<font color="Olive">a</font>,<font color="Olive">b</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 ) </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>);<br/>
<a NAME="E5:27"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E77</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:27_1_1"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 2
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
<br/>.= 
<font color="Maroon">m</font>

;<br/><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>
<br/>.= 
<font color="Maroon">m</font>

;<br/></div><b>end;</b></div>
<a NAME="E2:27_1"/>
<font color="Maroon">H<sub>1</sub></font>(0) <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(0)
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">b</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="E3:27_1_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E4:27_1_2_1"/><i><font color="Green">E83</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:27_1_2_1_1"/><i><font color="Green">E84</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 )
 ;<br/><a NAME="E2:27_1_2_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:27_1_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><a NAME="E6:27_1_2_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:27_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:27_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:27_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:27_1_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> 0</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:27_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(0)
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:27_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:27"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">w</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">w</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">w</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:27_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">w</font>]
 ;<br/>

<a NAME="E2:27_2"/><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E3:27_2"/><i><font color="Green">E89</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 3
 
<b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/>
<a NAME="E4:27_2"/><i><font color="Green">E90</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> 3
 
<b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/>
<a NAME="E5:27_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">w</font>) holds <br/> ex <font color="Olive">K</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1) &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Olive">K</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:27_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">w</font>)
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">b</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="E3:27_2_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span></span>)</span>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E4:27_2_1"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E5:27_2_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>and </b><br/><a NAME="E6:27_2_1"/><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E7:27_2_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:27_2_1_1"/>
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:27_2_1_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E3:27_2_1_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><a NAME="E4:27_2_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><b>hence </b><a NAME="E5:27_2_1_1"/>
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:27_2_1"/><b>then </b><i><font color="Green">E96</font></i>: 
( 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:27_2_1"/><b>then </b><i><font color="Green">E97</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:27_2_1"/><i><font color="Green">E99</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:27_2_1"/><i><font color="Green">E100</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:27_2_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E14:27_2_1"/><i><font color="Green">E102</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="jordan1d.html#E2"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:27_2_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E16:27_2_1"/><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> 2
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:27_2_1"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> 2
 
<b>by </b><i><a class="txt" href="jordan1d.html#E2"><i><font color="Green">Lemma40</font></i></a>, </i>;<br/>
<a NAME="E18:27_2_1"/><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 2 <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">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan1d.html#E5"><i><font color="Green">Lemma42</font></i></a>, , <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E19:27_2_1"/><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 2 <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">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, <a class="txt" href="jordan1d.html#E6"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<b>take </b><font color="Maroon">K</font> = <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="enumset1.html#K2">}</a></span>;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>
<div class="add"><b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_2_1_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><div><i><font color="Green">E109</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:27_2_1_2_1"/><i><font color="Green">E110</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><b>assume </b><a NAME="E2:27_2_1_2_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 ;<br/><a NAME="E3:27_2_1_2_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</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="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E4:27_2_1_2_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2
 ;<br/><a NAME="E5:27_2_1_2_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2
 <b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><a NAME="E6:27_2_1_2_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2
 <b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><a NAME="E7:27_2_1_2_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;=</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="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_2_1_2_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 2 <a href="xxreal_0.html#R1">&lt;=</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="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 2
 <b>by </b><i><a class="ref" href="jordan1d.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E9:27_2_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E10:27_2_1_2_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E112</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:27_2_1_2_2"/><i><font color="Green">E113</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><b>assume </b><a NAME="E2:27_2_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E3:27_2_1_2_2"/><b>then </b>
2 <a href="real_1.html#K4">*</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="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E4:27_2_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 4 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E5:27_2_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 4 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><a NAME="E6:27_2_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 4 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><a NAME="E7:27_2_1_2_2"/><b>then </b>
<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="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 4</span>)</span> <a href="real_1.html#K5">-</a> 2 <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">a</font></span>)</span> <a href="real_1.html#K5">-</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E8:27_2_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K5">-</a> 2</span>)</span> <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">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 2
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E4:27_2_1_2"/><b>then </b><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <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">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 2
 <b>by </b><i>, </i>;<br/><a NAME="E5:27_2_1_2"/><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, , <a class="txt" href="jordan1d.html#E3:27"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E6:27_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E7:27_2_1_2"/><b>then </b><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="txt" href="jordan1d.html#E4"><i><font color="Green">Lemma41</font></i></a>, </i>;<br/><a NAME="E8:27_2_1_2"/><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <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">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 2
 <b>by </b><i>, <a class="txt" href="jordan1d.html#E2"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="jordan1d.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E9:27_2_1_2"/><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="txt" href="jordan1d.html#E2"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="jordan1d.html#E3:27"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E10:27_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E11:27_2_1_2"/><b>then </b><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E12:27_2_1_2"/><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <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">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:27_2_1_2"/><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <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">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan1d.html#E1:27"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E14:27_2_1_2"/>
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span> or <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span> or <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> or <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan1d.html#E2:27"><i><font color="Green">E49</font></i></a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><b>hence </b><a NAME="E15:27_2_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1)
 <b>by </b><i>, <a class="txt" href="jordan1d.html#E4:27"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a>, , , , <a class="ref" href="jordan1d.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan1d.html#T3" target="_self">Th3</a></i>;<br/></div>
<b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:27_2_1_3"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/><a NAME="E2:27_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> 3</span>)</span> <a href="real_1.html#K5">-</a> 2
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E3:27_2_1_3"/><b>then </b>
<font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K5">-</a> 2</span>)</span>
 ;<br/><a NAME="E4:27_2_1_3"/><b>then </b>
<font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> 0
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E5:27_2_1_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E6:27_2_1_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>;<br/><a NAME="E7:27_2_1_3"/><b>then </b><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 3 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 3
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>assume </b><a NAME="E8:27_2_1_3"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 ;<br/><a NAME="E9:27_2_1_3"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</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="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E10:27_2_1_3"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</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"><span class="p5">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="sppol_1.html#T5">SPPOL_1:5</a></i>;<br/><b>hence </b><a NAME="E11:27_2_1_3"/>
<font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan1d.html#E2:27"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E22:27_2_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span></span>)</span> &amp; <font color="Maroon">b</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E23:27_2_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span></span>)</span>,<font color="Maroon">a</font>,<font color="Maroon">b</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"><a href="goboard5.html#K3">cell</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan1d.html#E2"><i><font color="Green">Lemma40</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E24:27_2_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">K</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:27_2"/>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1)
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<a NAME="E7:27_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1) <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">w</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a><br/>

<b>assume </b><a NAME="E1:27_2_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1)
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">b</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="E3:27_2_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E4:27_2_2"/><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E5:27_2_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>and </b><br/><a NAME="E6:27_2_2"/><i><font color="Green">E126</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E7:27_2_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 ;<br/>
<div><i><font color="Green">E128</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:27_2_2_1"/><i><font color="Green">E129</font></i>: 
 ex <font color="Olive">e</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Olive">e</font>
 <b>by </b><i><a class="ref" href="abian.html#D2">ABIAN:def 2</a></i>;<br/><b>thus </b>2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>

<br/>.= 
<font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i>, <a class="ref" href="nat_1.html#T72">NAT_1:72</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E130</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>consider </b><font color="Maroon">e</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="E2:27_2_2_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">e</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="abian.html#T9">ABIAN:9</a></i>;<br/><a NAME="E3:27_2_2_2"/><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">e</font></span>)</span> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T63">NAT_1:63</a></i>;<br/><b>thus </b>2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">e</font></span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i><a class="txt" href="jordan1d.html#E2"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="jordan1d.html#E4"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="nat_1.html#T69">NAT_1:69</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="nat_1.html#T72">NAT_1:72</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="jordan1d.html#E2"><i><font color="Green">Lemma40</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <a href="zfmisc_1.html#K1">K10</a>(the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>) =  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">a<sub>2</sub></font></span>)</span>;<br/>
<div><i><font color="Green">E133</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:27_2_2_3"/><i><font color="Green">E134</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 2
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
;<br/><a NAME="E3:27_2_2_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan1d.html#E4"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E4:27_2_2_3"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><b>hence </b><a NAME="E5:27_2_2_3"/>
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:27_2_2"/><b>then </b>
2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2
 
<b>by </b><i/>;<br/>
<a NAME="E12:27_2_2"/><b>then </b><i><font color="Green">E135</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:27_2_2"/>
2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2
 
<b>by </b><i>, <a class="txt" href="jordan1d.html#E2"><i><font color="Green">Lemma40</font></i></a></i>;<br/>
<a NAME="E14:27_2_2"/><b>then </b><i><font color="Green">E136</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>take </b><font color="Maroon">Y</font> =  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>

<a NAME="E15:27_2_2"/>
2 <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2
 
<b>by </b><i><a class="txt" href="jordan1d.html#E4"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="nat_2.html#T26">NAT_2:26</a></i>;<br/>
<a NAME="E16:27_2_2"/><b>then </b><i><font color="Green">E137</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <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/>
<div><i><font color="Green">E138</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b>2 <a href="real_1.html#K4">*</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="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 2</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<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="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 2

;<br/></div><b>end;</b></div>
<div><i><font color="Green">E139</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b>2 <a href="real_1.html#K4">*</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="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<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="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1

;<br/></div><b>end;</b></div>
<div><i><font color="Green">E140</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:27_2_2_6"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/><b>assume </b><a NAME="E2:27_2_2_6"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 ;<br/><a NAME="E3:27_2_2_6"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 3 <a href="xxreal_0.html#R1">&lt;=</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="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 3
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:27_2_2_6"/><b>then </b><i><font color="Green">E142</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</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"><span class="p5">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p0">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 3</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E5:27_2_2_6"/><b>then </b><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</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"><span class="p5">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p0">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 3</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E6:27_2_2_6"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> 3
 <b>by </b><i>, <a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/><a NAME="E7:27_2_2_6"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 3
 <b>by </b><i/>;<br/><a NAME="E8:27_2_2_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 3
 ;<br/><a NAME="E9:27_2_2_6"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 3
 <b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E10:27_2_2_6"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 3</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E11:27_2_2_6"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 3</span>)</span>
 ;<br/><a NAME="E12:27_2_2_6"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 3</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E13:27_2_2_6"/>
<span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E14:27_2_2_6"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> 2
 <b>by </b><i><a class="txt" href="jordan1d.html#E3:27"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E15:27_2_2_6"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> 2
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E16:27_2_2_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E17:27_2_2_6"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E18:27_2_2_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 5 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 6
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E19:27_2_2_6"/><b>then </b><i><font color="Green">E145</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 5 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 6
 <b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>;<br/><a NAME="E20:27_2_2_6"/><b>then </b><i><font color="Green">E146</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 6
 <b>by </b><i>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><a NAME="E21:27_2_2_6"/><i><font color="Green">E147</font></i>: 
<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="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 3</span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 3</span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan1d.html#T1" target="_self">Th1</a>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_2_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_2_2_6_1_1"/>
(  not <font color="Maroon">a</font> is <a href="abian.html#V1">even</a> or <font color="Maroon">a</font> is <a href="abian.html#V1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_2_6_1_1_1"><b>suppose </b></a><a NAME="E1:27_2_2_6_1_1_1"/><i><font color="Green">E148</font></i>: 
not <font color="Maroon">a</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add">2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>

;<br/><b>hence </b><a NAME="E3:27_2_2_6_1_1_1"/>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 3</span>)</span>
 <b>by </b><i><a class="txt" href="jordan1d.html#E1:27"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_2_6_1_1_2"><b>suppose </b></a><a NAME="E1:27_2_2_6_1_1_2"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">a</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add">2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 2</span>)</span>

;<br/><b>hence </b><a NAME="E3:27_2_2_6_1_1_2"/>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 3</span>)</span>
 <b>by </b><i><a class="txt" href="jordan1d.html#E2:27"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan1d.html#E4:27"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E23:27_2_2_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/></div><b>end;</b></div>
<a NAME="E20:27_2_2"/><b>then </b><i><font color="Green">E150</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E21:27_2_2"/>
2 <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> 2
 
<b>by </b><i><a class="txt" href="jordan1d.html#E5"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="nat_2.html#T26">NAT_2:26</a></i>;<br/>
<a NAME="E22:27_2_2"/><b>then </b><i><font color="Green">E151</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <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="E23:27_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="jordan1d.html#E6"><i><font color="Green">Lemma46</font></i></a></i>;<br/>
<a NAME="E24:27_2_2"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(2,2) <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(1,2)</span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(2,1)</span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(1,1)
 
<b>by </b><i>, , <a class="txt" href="jordan1d.html#E1:27"><i><font color="Green">E48</font></i></a>, </i>;<br/>
<div><i><font color="Green">E153</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:27_2_2_7"/><i><font color="Green">E154</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:27_2_2_7"/><i><font color="Green">E155</font></i>: 
2 <a href="real_1.html#K4">*</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="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span> <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="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 2
 <b>by </b><i/>;<br/><a NAME="E3:27_2_2_7"/>
<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="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="jordan1d.html#T1" target="_self">Th1</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:27_2_2_7"/><b>then </b>
2 <a href="real_1.html#K4">*</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="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:27_2_2_7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E156</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:27_2_2_8"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 ;<br/><a NAME="E2:27_2_2_8"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:27_2_2_8"/><i><font color="Green">E162</font></i>: 
2 <a href="real_1.html#K4">*</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="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <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="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E4:27_2_2_8"/>
<font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E5:27_2_2_8"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</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="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:27_2_2_8"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</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="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 ;<br/><a NAME="E7:27_2_2_8"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</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="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan1d.html#E4:27"><i><font color="Green">E53</font></i></a></i>;<br/><b>hence </b><a NAME="E8:27_2_2_8"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E163</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:27_2_2_9"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:27_2_2_9"/><b>then </b><i><font color="Green">E164</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:27_2_2_9"/><i><font color="Green">E165</font></i>: 
2 <a href="real_1.html#K4">*</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="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span> <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="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 2
 <b>by </b><i/>;<br/><a NAME="E4:27_2_2_9"/>
<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="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan1d.html#E4:27"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:27_2_2_9"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p0">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E6:27_2_2_9"/><b>then </b>
2 <a href="real_1.html#K4">*</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="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>;<br/><b>hence </b><a NAME="E7:27_2_2_9"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E166</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:27_2_2_10"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 ;<br/><a NAME="E2:27_2_2_10"/><i><font color="Green">E168</font></i>: 
2 <a href="real_1.html#K4">*</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="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <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="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E3:27_2_2_10"/>
<font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</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="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:27_2_2_10"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</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="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:27_2_2_10"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_2_11"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_2_2_11"/>
( ( not <font color="Maroon">a</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">b</font> is <a href="abian.html#V1">even</a> ) or ( not <font color="Maroon">a</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">b</font> is <a href="abian.html#V1">even</a> ) or ( <font color="Maroon">a</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">b</font> is <a href="abian.html#V1">even</a> ) or ( <font color="Maroon">a</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">b</font> is <a href="abian.html#V1">even</a> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_2_11_1"><b>suppose </b></a><a NAME="E1:27_2_2_11_1"/><i><font color="Green">E169</font></i>: 
( not <font color="Maroon">a</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">b</font> is <a href="abian.html#V1">even</a> )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E170</font></i>: <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/><a NAME="E3:27_2_2_11_1"/><i><font color="Green">E171</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E4:27_2_2_11_1"/><i><font color="Green">E172</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="txt" href="jordan1d.html#E4:27"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E5:27_2_2_11_1"/><i><font color="Green">E173</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E6:27_2_2_11_1"/>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="txt" href="jordan1d.html#E4:27"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>;<br/><b>hence </b><a NAME="E7:27_2_2_11_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">w</font>)
 <b>by </b><i>, , <a class="ref" href="jordan1d.html#T2" target="_self">Th2</a></i>;<br/><span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/><b>hence </b><a NAME="E9:27_2_2_11_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="txt" href="jordan1d.html#E2:27"><i><font color="Green">E49</font></i></a>, , <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_2_11_2"><b>suppose </b></a><a NAME="E1:27_2_2_11_2"/><i><font color="Green">E174</font></i>: 
( not <font color="Maroon">a</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">b</font> is <a href="abian.html#V1">even</a> )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E175</font></i>: <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/><i><font color="Green">E176</font></i>: <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 2
<b>by </b><i>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/><a NAME="E4:27_2_2_11_2"/><i><font color="Green">E177</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E5:27_2_2_11_2"/><i><font color="Green">E178</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="txt" href="jordan1d.html#E4:27"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E6:27_2_2_11_2"/><i><font color="Green">E179</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="jordan1d.html#E3:27"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E7:27_2_2_11_2"/>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="ref" href="jordan1d.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>;<br/><b>hence </b><a NAME="E8:27_2_2_11_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">w</font>)
 <b>by </b><i>, <a class="ref" href="jordan1d.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan1d.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E9:27_2_2_11_2"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(1,2) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>2</sub></font>(2,2) <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(1,2)
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E10:27_2_2_11_2"/>
<font color="Maroon">H<sub>2</sub></font>(2,2) <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(1,2) <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(2,2) <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(1,2)</span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(2,1)
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E11:27_2_2_11_2"/><b>then </b><i><font color="Green">E181</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(1,2) <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(2,2) <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(1,2)</span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(2,1)
 <b>by </b><i><a class="ref" href="jordan1d.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:27_2_2_11_2"/>
<span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(2,2) <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(1,2)</span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(2,1) <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="jordan1d.html#E2:27"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E13:27_2_2_11_2"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , , <a class="txt" href="jordan1d.html#E13"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_2_11_3"><b>suppose </b></a><a NAME="E1:27_2_2_11_3"/><i><font color="Green">E182</font></i>: 
( <font color="Maroon">a</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">b</font> is <a href="abian.html#V1">even</a> )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E183</font></i>: <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 2
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/><i><font color="Green">E184</font></i>: <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/><a NAME="E4:27_2_2_11_3"/><i><font color="Green">E185</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="jordan1d.html#E3:27"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E5:27_2_2_11_3"/><i><font color="Green">E186</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="ref" href="jordan1d.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E6:27_2_2_11_3"/><i><font color="Green">E187</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E7:27_2_2_11_3"/>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="txt" href="jordan1d.html#E4:27"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>;<br/><b>hence </b><a NAME="E8:27_2_2_11_3"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">w</font>)
 <b>by </b><i>, <a class="ref" href="jordan1d.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan1d.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E9:27_2_2_11_3"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(2,1) <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(2,2) <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(1,2)</span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(2,1)
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E10:27_2_2_11_3"/>
<span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(2,2) <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(1,2)</span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(2,1) <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="jordan1d.html#E2:27"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E11:27_2_2_11_3"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , , <a class="ref" href="jordan1d.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_2_11_4"><b>suppose </b></a><a NAME="E1:27_2_2_11_4"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">a</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">b</font> is <a href="abian.html#V1">even</a> )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E190</font></i>: <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 2
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/><a NAME="E3:27_2_2_11_4"/><i><font color="Green">E191</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="jordan1d.html#E3:27"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E4:27_2_2_11_4"/><i><font color="Green">E192</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="ref" href="jordan1d.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E5:27_2_2_11_4"/><i><font color="Green">E193</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="jordan1d.html#E3:27"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E6:27_2_2_11_4"/>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="ref" href="jordan1d.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>;<br/><b>hence </b><a NAME="E7:27_2_2_11_4"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">w</font>)
 <b>by </b><i>, , <a class="ref" href="jordan1d.html#T2" target="_self">Th2</a></i>;<br/><span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 2
<b>by </b><i>, <a class="txt" href="jordan1d.html#E5:27"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/><a NAME="E9:27_2_2_11_4"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>2</sub></font>(2,2) <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(1,2) <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(2,1)</span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(1,1)</span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E10:27_2_2_11_4"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(2,2) <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(1,2) <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(2,1)</span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H<sub>2</sub></font>(1,1)
 <b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><b>hence </b><a NAME="E11:27_2_2_11_4"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="jordan1d.html#E2:27"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E8:27_2"/><b>then </b><i><font color="Green">E194</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1) <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">w</font>)
 
<b>by </b><i><a class="ref" href="setfam_1.html#T18">SETFAM_1:18</a></i>;<br/>
<b>let </b><font color="Maroon">d</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E9:27_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1)
 ;<br/>

<b>hence </b><a NAME="E10:27_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:27"/>
for <font color="Olive">w</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">w</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E8:27"/>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">D</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Olive">a</font>,<font color="Olive">b</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 ) </span>}</span> 
 ;<br/>


</div>
