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

<b>set </b><font color="Maroon">DD</font> = <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/>( ( for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">r</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) &amp;  ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">s</font> ) ) </span>}</span> ;<br/>
<a NAME="E1:19"/>
0 <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">0'</font> = 0 as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i><a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">a<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>] </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>consider </b><font color="Maroon">half</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E5:19"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T61">ARYTM_3:61</a></i>;<br/>
<a NAME="E6:19"/><i><font color="Green">E25</font></i>: 
 <a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">two</font>
 
<b>by </b><i><a class="txt" href="numbers.html#E5:19"><i><font color="Green">E13</font></i></a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/>
<a NAME="E7:19"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/>
<a NAME="E8:19"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">r</font>, <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">t</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">r</font> holds <br/><font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> ;<br/><b>let </b><font color="Maroon">t</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> ;<br/>



<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:19_1"/><b>then </b><i><font color="Green">E30</font></i>: 
 ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font> )
 
;<br/>
<b>assume </b><a NAME="E3:19_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">t</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E4:19_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">t</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T90">ARYTM_3:90</a></i>;<br/>
<a NAME="E5:19_1"/>
<font color="Maroon">t</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="arytm_3.html#T90">ARYTM_3:90</a></i>;<br/>
<a NAME="E6:19_1"/><b>then </b>
<font color="Maroon">t</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="numbers.html#E9"><i><font color="Green">Lemma22</font></i></a>, <a class="ref" href="arytm_3.html#T74">ARYTM_3:74</a></i>;<br/>
<a NAME="E7:19_1"/><b>then </b>
<font color="Maroon">t</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font>
 
<b>by </b><i>, <a class="ref" href="arytm_3.html#T76">ARYTM_3:76</a></i>;<br/>
<b>hence </b><a NAME="E8:19_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:19"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">1'</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">1'</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>;<br/>
2 = 
 <a href="ordinal1.html#K1">succ</a> 1

<br/>.= 
1 <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="ordinal1.html#D1">ORDINAL1:def 1</a></i>
;<br/>
<a NAME="E11:19"/><b>then </b>
1 <a href="ordinal1.html#R1">c=</a> 2
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E12:19"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">1'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">two</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:19"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">1'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/>
<a NAME="E14:19"/><b>then </b><i><font color="Green">E36</font></i>: 
1 <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:19"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">0'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">1'</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T71">ARYTM_3:71</a></i>;<br/>
<a NAME="E16:19"/><b>then </b><i><font color="Green">E38</font></i>: 
0 <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:19"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">1'</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font> <a href="hidden.html#R1">=</a> <font color="Maroon">half</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>;<br/>
<a NAME="E18:19"/>
<font color="Maroon">1'</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font>
 
<b>by </b><i>, <a class="ref" href="arytm_3.html#T90">ARYTM_3:90</a></i>;<br/>
<a NAME="E19:19"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">half</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="numbers.html#E8"><i><font color="Green">Lemma16</font></i></a>, , , <a class="txt" href="numbers.html#E10"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_3"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><b>then </b><font color="Maroon">X</font> = 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,0</span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="tarski.html#D5">TARSKI:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>
;<br/><b>hence </b><a NAME="E3:19_3"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E6:19"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:19"/><i><font color="Green">E46</font></i>: 
2 <a href="arytm_3.html#K3">*^</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">two</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T65">ARYTM_3:65</a></i>;<br/>
<a NAME="E22:19"/><i><font color="Green">E47</font></i>: 
1 <a href="arytm_3.html#K3">*^</a> 2 <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>;<br/>
<a NAME="E23:19"/>
2 <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> 1
 
;<br/>
<a NAME="E24:19"/><b>then </b>
1 <a href="hidden.html#R2">in</a> 2
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E25:19"/><b>then </b><i><font color="Green">E48</font></i>: 
1 <a href="arytm_3.html#K3">*^</a> 2 <a href="hidden.html#R2">in</a> 2 <a href="arytm_3.html#K3">*^</a> 2
 
<b>by </b><i><a class="ref" href="ordinal3.html#T22">ORDINAL3:22</a></i>;<br/>
<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_4"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">t</font> </span>}</span>  where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Olive">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">t0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E3:19_4"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t0</font> </span>}</span> 
 <b>and </b><br/><a NAME="E4:19_4"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">t0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><b>set </b><font color="Maroon">n</font> =  <a href="arytm_3.html#K8">numerator</a> <font color="Maroon">t0</font>;<br/><b>set </b><font color="Maroon">d</font> =  <a href="arytm_3.html#K9">denominator</a> <font color="Maroon">t0</font>;<br/><div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_4_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">two</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_4_1_1"/>
( <font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font> or <font color="Maroon">two</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> )
 </span><b>by </b><i><a class="txt" href="numbers.html#E14:19"><i><font color="Green">E36</font></i></a>, <a class="ref" href="arytm_3.html#T73">ARYTM_3:73</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1_1"><b>suppose </b></a><a NAME="E1:19_4_1_1_1"/>
<font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font>
 ;<br/><div class="add"><a NAME="E2:19_4_1_1_1"/><b>then </b>
<font color="Maroon">t0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E3:19_4_1_1_1"/><b>then </b>
 ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t0</font> &amp; <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t0</font> )
 <b>by </b><i><a class="txt" href="numbers.html#E9:19"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E4:19_4_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1_2"><b>suppose </b></a><a NAME="E1:19_4_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">two</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">es</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E3:19_4_1_1_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">two</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">es</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> or <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">es</font> <a href="hidden.html#R1">=</a> <font color="Maroon">two</font> )
 <b>by </b><i><a class="ref" href="arytm_3.html#T100">ARYTM_3:100</a></i>;<br/><a NAME="E4:19_4_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">0'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">es</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T71">ARYTM_3:71</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_4_1_1_2_1"/>
<font color="Maroon">0'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">es</font>
 ;<br/><a NAME="E2:19_4_1_1_2_1"/><b>then </b>
<font color="Maroon">two</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">es</font> <a href="hidden.html#R1">=</a> <font color="Maroon">two</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T56">ARYTM_3:56</a></i>;<br/><b>hence </b><a NAME="E3:19_4_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E15:19"><i><font color="Green">E37</font></i></a>, <a class="txt" href="numbers.html#E16:19"><i><font color="Green">E38</font></i></a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/></div><b>end;</b></div><a NAME="E6:19_4_1_1_2"/><b>then </b>
<font color="Maroon">0'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">es</font>
 <b>by </b><i><a class="txt" href="numbers.html#E17:19"><i><font color="Green">E39</font></i></a>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E8:19_4_1_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">0'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E9:19_4_1_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">es</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T101">ARYTM_3:101</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_4_1_1_2_2_1"/>
( <font color="Maroon">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14">one</a>  or  <a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:19_4_1_1_2_2_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14">one</a> 
 ;<br/><div class="add"><a NAME="E2:19_4_1_1_2_2_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="numbers.html#E19:19"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E3:19_4_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T88">ARYTM_3:88</a></i>;<br/><a NAME="E4:19_4_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>;<br/><b>consider </b><font color="Maroon">t02s2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E6:19_4_1_1_2_2_1_1"/><i><font color="Green">E91</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t02s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> or <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t02s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="arytm_3.html#T100">ARYTM_3:100</a></i>;<br/><a NAME="E7:19_4_1_1_2_2_1_1"/>
<font color="Maroon">es</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="txt" href="numbers.html#E15:19"><i><font color="Green">E37</font></i></a>, <a class="txt" href="numbers.html#E16:19"><i><font color="Green">E38</font></i></a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/><a NAME="E8:19_4_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="txt" href="numbers.html#E20:19"><i><font color="Green">E41</font></i></a>, <a class="ref" href="arytm_3.html#T76">ARYTM_3:76</a></i>;<br/><a NAME="E9:19_4_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T77">ARYTM_3:77</a></i>;<br/><a NAME="E10:19_4_1_1_2_2_1_1"/>
<font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">es</font>
 <b>by </b><i><a class="txt" href="numbers.html#E20:19"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="arytm_3.html#T77">ARYTM_3:77</a></i>;<br/><a NAME="E11:19_4_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">two</font> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">es</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T83">ARYTM_3:83</a></i>;<br/><a NAME="E12:19_4_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">two</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t02s2</font>
 <b>by </b><i><a class="txt" href="numbers.html#E15:19"><i><font color="Green">E37</font></i></a>, <a class="txt" href="numbers.html#E16:19"><i><font color="Green">E38</font></i></a>, , <a class="txt" href="numbers.html#E21:19"><i><font color="Green">E46</font></i></a>, <a class="ref" href="arytm_3.html#T83">ARYTM_3:83</a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/><a NAME="E13:19_4_1_1_2_2_1_1"/>
<font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="numbers.html#E12:19"><i><font color="Green">E34</font></i></a>, <a class="ref" href="arytm_3.html#T86">ARYTM_3:86</a></i>;<br/><b>then consider </b><font color="Maroon">2t'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E15:19_4_1_1_2_2_1_1"/><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T61">ARYTM_3:61</a></i>;<br/><b>set </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font>;<br/><b>consider </b><font color="Maroon">t0x</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E17:19_4_1_1_2_2_1_1"/><i><font color="Green">E97</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t0x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t0</font> or <font color="Maroon">t0</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t0x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font> )
 <b>by </b><i><a class="ref" href="arytm_3.html#T100">ARYTM_3:100</a></i>;<br/><a NAME="E18:19_4_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i><a class="txt" href="numbers.html#E25:19"><i><font color="Green">E48</font></i></a>, <a class="ref" href="arytm_3.html#T58">ARYTM_3:58</a></i>;<br/><a NAME="E19:19_4_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E98</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font> or <font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="arytm_3.html#R3">&lt;='</a>  <a href="arytm_3.html#K14">one</a>  )
 <b>by </b><i><a class="ref" href="arytm_3.html#T91">ARYTM_3:91</a></i>;<br/><div><i><font color="Green">E99</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_2_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_4_1_1_2_2_1_1_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">t0</font> <a href="arytm_3.html#R3">&lt;='</a>  <a href="arytm_3.html#K14">one</a> 
 ;<br/><a NAME="E2:19_4_1_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T90">ARYTM_3:90</a></i>;<br/><a NAME="E3:19_4_1_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>;<br/><a NAME="E4:19_4_1_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="arytm_3.html#R3">&lt;='</a>  <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T74">ARYTM_3:74</a></i>;<br/><b>hence </b><a NAME="E5:19_4_1_1_2_2_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="numbers.html#E15:19"><i><font color="Green">E37</font></i></a>, <a class="ref" href="arytm_3.html#T76">ARYTM_3:76</a></i>;<br/></div><b>end;</b></div><a NAME="E21:19_4_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#NR4" target="_self">&lt;</a> <a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T88">ARYTM_3:88</a></i>;<br/><a NAME="E22:19_4_1_1_2_2_1_1"/>
<a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i>, <a class="txt" href="numbers.html#E12:19"><i><font color="Green">E34</font></i></a>, <a class="ref" href="arytm_3.html#T88">ARYTM_3:88</a></i>;<br/><a NAME="E23:19_4_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T77">ARYTM_3:77</a></i>;<br/><a NAME="E24:19_4_1_1_2_2_1_1"/>
<font color="Maroon">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t0</font>
 <b>by </b><i>, , <a class="ref" href="arytm_3.html#T77">ARYTM_3:77</a></i>;<br/><a NAME="E25:19_4_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t0</font>
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T77">ARYTM_3:77</a></i>;<br/><a NAME="E26:19_4_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t0</font>
 <b>by </b><i>, <a class="txt" href="numbers.html#E3:19_4"><i><font color="Green">E53</font></i></a>, <a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a>, <a class="ref" href="arytm_3.html#T76">ARYTM_3:76</a></i>;<br/><div><i><font color="Green">E104</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_2_2_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_4_1_1_2_2_1_1_2"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1_2_2_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_4_1_1_2_2_1_1_2_1"/>
( <font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">2t'</font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i><a class="txt" href="numbers.html#E1:19_4_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="arytm_3.html#T86">ARYTM_3:86</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1_2_2_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:19_4_1_1_2_2_1_1_2_1_1"/>
<font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_4_1_1_2_2_1_1_2_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T86">ARYTM_3:86</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1_2_2_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:19_4_1_1_2_2_1_1_2_1_2"/>
<font color="Maroon">2t'</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_4_1_1_2_2_1_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E25:19"><i><font color="Green">E48</font></i></a>, <a class="ref" href="arytm_3.html#T54">ARYTM_3:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green">E106</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"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T57">ARYTM_3:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span>
<b>by </b><i><a class="txt" href="numbers.html#E26:19"><i><font color="Green">E49</font></i></a>, <a class="txt" href="numbers.html#E4:19_4"><i><font color="Green">E54</font></i></a>, <a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</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"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> </span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</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"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> </span>)</span> <a href="arytm_3.html#K11">+</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"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> </span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">two</font>
<b>by </b><i><a class="txt" href="numbers.html#E5:19"><i><font color="Green">E13</font></i></a>, <a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">two</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T58">ARYTM_3:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> 
<b>by </b><i><a class="txt" href="numbers.html#E25:19"><i><font color="Green">E48</font></i></a>, <a class="ref" href="arytm_3.html#T58">ARYTM_3:58</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">t02s2</font> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0x</font> <a href="arytm_3.html#K11">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="numbers.html#E21:19"><i><font color="Green">E46</font></i></a>, <a class="txt" href="numbers.html#E26:19"><i><font color="Green">E49</font></i></a>, <a class="txt" href="numbers.html#E4:19_4"><i><font color="Green">E54</font></i></a>, <a class="ref" href="arytm_3.html#T57">ARYTM_3:57</a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0x</font> <a href="arytm_3.html#K12">*'</a> <span class="p3">(<span class="default"><font color="Maroon">t0x</font> <a href="arytm_3.html#K11">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numbers.html#E26:19"><i><font color="Green">E49</font></i></a>, <a class="txt" href="numbers.html#E4:19_4"><i><font color="Green">E54</font></i></a>, <a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t0x</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0x</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</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"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T57">ARYTM_3:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t0x</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span> <a href="arytm_3.html#K11">+</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"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T57">ARYTM_3:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t0x</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="txt" href="numbers.html#E1:19_4_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="arytm_3.html#T57">ARYTM_3:57</a></i>
;<br/><a NAME="E30:19_4_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">t0x</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t02s2</font> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="arytm_3.html#T69">ARYTM_3:69</a></i>;<br/><a NAME="E31:19_4_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">t02s2</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t0x</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/><a NAME="E32:19_4_1_1_2_2_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">t0x</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="txt" href="numbers.html#E26:19"><i><font color="Green">E49</font></i></a>, <a class="txt" href="numbers.html#E4:19_4"><i><font color="Green">E54</font></i></a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/><a NAME="E33:19_4_1_1_2_2_1_1"/>
<font color="Maroon">t0x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="txt" href="numbers.html#E26:19"><i><font color="Green">E49</font></i></a>, <a class="txt" href="numbers.html#E4:19_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="numbers.html#E5:19_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="arytm_3.html#T92">ARYTM_3:92</a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/><a NAME="E34:19_4_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">t0x</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="txt" href="numbers.html#E3:19_4_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/><a NAME="E35:19_4_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">t0x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="numbers.html#E9:19"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E36:19_4_1_1_2_2_1_1"/><b>then </b>
 ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t0x</font> &amp; <font color="Olive">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font> )
 ;<br/><b>hence </b><a NAME="E37:19_4_1_1_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E22:19"><i><font color="Green">E47</font></i></a>, <a class="txt" href="numbers.html#E1:19_4_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="arytm_3.html#T76">ARYTM_3:76</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:19_4_1_1_2_2_1_2"/><i><font color="Green">E109</font></i>: 
 <a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:19_4_1_1_2_2_1_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">half</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i><a class="txt" href="numbers.html#E8"><i><font color="Green">Lemma16</font></i></a>, <a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>;<br/><a NAME="E3:19_4_1_1_2_2_1_2"/>
<font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">two</font> <a href="hidden.html#R1">=</a> <a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i><a class="txt" href="numbers.html#E8"><i><font color="Green">Lemma16</font></i></a>, <a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>;<br/><a NAME="E4:19_4_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">half</font> <a href="arytm_3.html#R3">&lt;='</a>  <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T91">ARYTM_3:91</a></i>;<br/><a NAME="E5:19_4_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">half</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/><a NAME="E6:19_4_1_1_2_2_1_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">half</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="numbers.html#E8"><i><font color="Green">Lemma16</font></i></a>, <a class="ref" href="arytm_3.html#T54">ARYTM_3:54</a></i>;<br/><a NAME="E7:19_4_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T88">ARYTM_3:88</a></i>;<br/><a NAME="E8:19_4_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">half</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>;<br/><a NAME="E9:19_4_1_1_2_2_1_2"/>
<font color="Maroon">half</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="arytm_3.html#T76">ARYTM_3:76</a></i>;<br/><a NAME="E10:19_4_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">half</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">es</font>
 <b>by </b><i><a class="txt" href="numbers.html#E20:19"><i><font color="Green">E41</font></i></a>, <a class="ref" href="arytm_3.html#T77">ARYTM_3:77</a></i>;<br/><b>set </b><font color="Maroon">s</font> = <font color="Maroon">half</font>;<br/><b>consider </b><font color="Maroon">t02s2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E12:19_4_1_1_2_2_1_2"/><i><font color="Green">E115</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t02s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> or <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t02s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font> )
 <b>by </b><i><a class="ref" href="arytm_3.html#T100">ARYTM_3:100</a></i>;<br/><a NAME="E13:19_4_1_1_2_2_1_2"/>
<font color="Maroon">es</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="txt" href="numbers.html#E15:19"><i><font color="Green">E37</font></i></a>, <a class="txt" href="numbers.html#E16:19"><i><font color="Green">E38</font></i></a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/><a NAME="E14:19_4_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">half</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="txt" href="numbers.html#E22:19"><i><font color="Green">E47</font></i></a>, <a class="ref" href="arytm_3.html#T76">ARYTM_3:76</a></i>;<br/><a NAME="E15:19_4_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="txt" href="numbers.html#E21:19"><i><font color="Green">E46</font></i></a>, <a class="ref" href="arytm_3.html#T77">ARYTM_3:77</a></i>;<br/><a NAME="E16:19_4_1_1_2_2_1_2"/>
<font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">es</font>
 <b>by </b><i><a class="txt" href="numbers.html#E21:19"><i><font color="Green">E46</font></i></a>, <a class="txt" href="numbers.html#E22:19"><i><font color="Green">E47</font></i></a>, <a class="ref" href="arytm_3.html#T77">ARYTM_3:77</a></i>;<br/><a NAME="E17:19_4_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">two</font> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">es</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T83">ARYTM_3:83</a></i>;<br/><a NAME="E18:19_4_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">two</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t02s2</font>
 <b>by </b><i><a class="txt" href="numbers.html#E15:19"><i><font color="Green">E37</font></i></a>, <a class="txt" href="numbers.html#E16:19"><i><font color="Green">E38</font></i></a>, <a class="txt" href="numbers.html#E25:19"><i><font color="Green">E48</font></i></a>, <a class="txt" href="numbers.html#E26:19"><i><font color="Green">E49</font></i></a>, <a class="ref" href="arytm_3.html#T83">ARYTM_3:83</a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/><a NAME="E19:19_4_1_1_2_2_1_2"/>
<font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="numbers.html#E12:19"><i><font color="Green">E34</font></i></a>, <a class="ref" href="arytm_3.html#T86">ARYTM_3:86</a></i>;<br/><b>then consider </b><font color="Maroon">2t'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E21:19_4_1_1_2_2_1_2"/><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T61">ARYTM_3:61</a></i>;<br/><b>set </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font>;<br/><b>consider </b><font color="Maroon">t0x</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E23:19_4_1_1_2_2_1_2"/><i><font color="Green">E119</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t0x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t0</font> or <font color="Maroon">t0</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t0x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font> )
 <b>by </b><i><a class="ref" href="arytm_3.html#T100">ARYTM_3:100</a></i>;<br/><a NAME="E24:19_4_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T58">ARYTM_3:58</a></i>;<br/><a NAME="E25:19_4_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font> or <font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="arytm_3.html#R3">&lt;='</a>  <a href="arytm_3.html#K14">one</a>  )
 <b>by </b><i><a class="ref" href="arytm_3.html#T91">ARYTM_3:91</a></i>;<br/><a NAME="E26:19_4_1_1_2_2_1_2"/>
 <a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">two</font>
 <b>by </b><i><a class="txt" href="numbers.html#E5:19"><i><font color="Green">E13</font></i></a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/><a NAME="E27:19_4_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E121</font></i>: 
 <a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/><div><i><font color="Green">E122</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_2_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_4_1_1_2_2_1_2_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">t0</font> <a href="arytm_3.html#R3">&lt;='</a>  <a href="arytm_3.html#K14">one</a> 
 ;<br/><a NAME="E2:19_4_1_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T90">ARYTM_3:90</a></i>;<br/><a NAME="E3:19_4_1_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>;<br/><a NAME="E4:19_4_1_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="arytm_3.html#R3">&lt;='</a>  <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i><a class="txt" href="numbers.html#E1:19_4_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="arytm_3.html#T74">ARYTM_3:74</a></i>;<br/><b>hence </b><a NAME="E5:19_4_1_1_2_2_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="numbers.html#E15:19"><i><font color="Green">E37</font></i></a>, <a class="ref" href="arytm_3.html#T76">ARYTM_3:76</a></i>;<br/></div><b>end;</b></div><a NAME="E29:19_4_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#NR4" target="_self">&lt;</a> <a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T88">ARYTM_3:88</a></i>;<br/><a NAME="E30:19_4_1_1_2_2_1_2"/>
<a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="txt" href="numbers.html#E12:19"><i><font color="Green">E34</font></i></a>, <a class="txt" href="numbers.html#E4:19_4"><i><font color="Green">E54</font></i></a>, <a class="ref" href="arytm_3.html#T88">ARYTM_3:88</a></i>;<br/><a NAME="E31:19_4_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E125</font></i>: 
<a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="txt" href="numbers.html#E1:19_4_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="arytm_3.html#T77">ARYTM_3:77</a></i>;<br/><a NAME="E32:19_4_1_1_2_2_1_2"/>
<font color="Maroon">half</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t0</font>
 <b>by </b><i>, <a class="txt" href="numbers.html#E5:19_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="arytm_3.html#T77">ARYTM_3:77</a></i>;<br/><a NAME="E33:19_4_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="txt" href="numbers.html#E21:19"><i><font color="Green">E46</font></i></a>, <a class="ref" href="arytm_3.html#T77">ARYTM_3:77</a></i>;<br/><a NAME="E34:19_4_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="txt" href="numbers.html#E3:19_4"><i><font color="Green">E53</font></i></a>, <a class="txt" href="numbers.html#E1:19_4_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a>, <a class="ref" href="arytm_3.html#T76">ARYTM_3:76</a></i>;<br/><div><i><font color="Green">E127</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1_1_2_2_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_4_1_1_2_2_1_2_2"/><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1_2_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_4_1_1_2_2_1_2_2_1"/>
( <font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">2t'</font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i><a class="ref" href="numbers.html#T1" target="_self">Th1</a>, <a class="ref" href="arytm_3.html#T86">ARYTM_3:86</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1_2_2_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:19_4_1_1_2_2_1_2_2_1_1"/>
<font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_4_1_1_2_2_1_2_2_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T86">ARYTM_3:86</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1_2_2_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:19_4_1_1_2_2_1_2_2_1_2"/>
<font color="Maroon">2t'</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_4_1_1_2_2_1_2_2_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T54">ARYTM_3:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green">E129</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"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T57">ARYTM_3:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span>
<b>by </b><i>, <a class="txt" href="numbers.html#E3:19_4_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</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"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> </span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</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"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> </span>)</span> <a href="arytm_3.html#K11">+</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"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> </span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">two</font>
<b>by </b><i><a class="txt" href="numbers.html#E5:19"><i><font color="Green">E13</font></i></a>, <a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">two</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T58">ARYTM_3:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> 
<b>by </b><i>, <a class="ref" href="arytm_3.html#T58">ARYTM_3:58</a></i>
<br/>.= 
<font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">t02s2</font> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numbers.html#E25:19"><i><font color="Green">E48</font></i></a>, <a class="txt" href="numbers.html#E26:19"><i><font color="Green">E49</font></i></a>, <a class="ref" href="arytm_3.html#T57">ARYTM_3:57</a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0x</font> <a href="arytm_3.html#K12">*'</a> <span class="p3">(<span class="default"><font color="Maroon">t0x</font> <a href="arytm_3.html#K11">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="numbers.html#E3:19_4_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t0x</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0x</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</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"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T57">ARYTM_3:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t0x</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span> <a href="arytm_3.html#K11">+</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"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T57">ARYTM_3:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t0x</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span>
<b>by </b><i><a class="ref" href="numbers.html#T1" target="_self">Th1</a>, <a class="ref" href="arytm_3.html#T57">ARYTM_3:57</a></i>
;<br/><a NAME="E38:19_4_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">t0x</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t02s2</font> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">half</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">2t'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="arytm_3.html#T69">ARYTM_3:69</a></i>;<br/><a NAME="E39:19_4_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">t02s2</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t0x</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0x</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/><a NAME="E40:19_4_1_1_2_2_1_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">t0x</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t0</font>
 <b>by </b><i>, <a class="txt" href="numbers.html#E3:19_4_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/><a NAME="E41:19_4_1_1_2_2_1_2"/>
<font color="Maroon">t0x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t0</font>
 <b>by </b><i>, <a class="txt" href="numbers.html#E3:19_4_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="numbers.html#E4:19_4_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="arytm_3.html#T92">ARYTM_3:92</a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/><a NAME="E42:19_4_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">t0x</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="txt" href="numbers.html#E12"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/><a NAME="E43:19_4_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">t0x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="numbers.html#E9:19"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E44:19_4_1_1_2_2_1_2"/><b>then </b>
 ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t0x</font> &amp; <font color="Olive">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font> )
 ;<br/><b>hence </b><a NAME="E45:19_4_1_1_2_2_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="arytm_3.html#T76">ARYTM_3:76</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E11:19_4_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E6:19_4"/>
 <a href="arytm_3.html#K9">denominator</a> <font color="Maroon">t0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="arytm_3.html#T41">ARYTM_3:41</a></i>;<br/><a NAME="E7:19_4"/><b>then </b><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">t0</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i><a class="ref" href="ordinal3.html#T34">ORDINAL3:34</a></i>;<br/><a NAME="E8:19_4"/>
<font color="Maroon">two</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">t0</font></span>)</span></span>)</span> <a href="arytm_3.html#K10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">t0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numbers.html#E13:19"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E9:19_4"/><b>then </b>
<font color="Maroon">two</font> <a href="arytm_3.html#K10">/</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">t0</font></span>)</span></span>)</span> <a href="arytm_3.html#K10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">t0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="arytm_3.html#T46">ARYTM_3:46</a></i>;<br/><b>then </b><i><font color="Green">E133</font></i>: <font color="Maroon">two</font> <a href="arytm_3.html#K3">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">t0</font></span>)</span></span>)</span> = 
1 <a href="arytm_3.html#K3">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">t0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numbers.html#E14:19"><i><font color="Green">E36</font></i></a>, <a class="ref" href="arytm_3.html#T51">ARYTM_3:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8">numerator</a> <font color="Maroon">t0</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>
;<br/><b>then consider </b><font color="Maroon">k</font> being  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E12:19_4"/><i><font color="Green">E134</font></i>: 
 <a href="arytm_3.html#K8">numerator</a> <font color="Maroon">t0</font> <a href="hidden.html#R1">=</a> 2 <a href="arytm_3.html#K3">*^</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="numbers.html#E3"><i><font color="Green">Lemma12</font></i></a></i>;<br/><a NAME="E13:19_4"/>
<font color="Maroon">two</font> <a href="arytm_3.html#K3">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">t0</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="arytm_3.html#K3">*^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="arytm_3.html#K3">*^</a> <span class="p2">(<span class="default">2 <a href="arytm_3.html#K3">*^</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numbers.html#E15:19"><i><font color="Green">E37</font></i></a>, <a class="txt" href="numbers.html#E16:19"><i><font color="Green">E38</font></i></a>, <a class="ref" href="ordinal3.html#T58">ORDINAL3:58</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">t0</font></span>)</span> <a href="arytm_3.html#K3">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9">denominator</a> <font color="Maroon">t0</font></span>)</span> = 
<font color="Maroon">k</font> <a href="arytm_3.html#K3">*^</a> <span class="p1">(<span class="default">2 <a href="arytm_3.html#K3">*^</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T36">ORDINAL3:36</a></i>
<br/>.= 
2 <a href="arytm_3.html#K3">*^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="arytm_3.html#K3">*^</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T58">ORDINAL3:58</a></i>
;<br/><b>then consider </b><font color="Maroon">p</font> being  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E16:19_4"/><i><font color="Green">E136</font></i>: 
 <a href="arytm_3.html#K9">denominator</a> <font color="Maroon">t0</font> <a href="hidden.html#R1">=</a> 2 <a href="arytm_3.html#K3">*^</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="numbers.html#E3"><i><font color="Green">Lemma12</font></i></a></i>;<br/><a NAME="E17:19_4"/>
 <a href="arytm_3.html#K8">numerator</a> <font color="Maroon">t0</font>, <a href="arytm_3.html#K9">denominator</a> <font color="Maroon">t0</font> <a href="arytm_3.html#R1">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T40">ARYTM_3:40</a></i>;<br/><b>hence </b><a NAME="E18:19_4"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E16:19"><i><font color="Green">E38</font></i></a>, <a class="txt" href="numbers.html#E17:19"><i><font color="Green">E39</font></i></a>, <a class="ref" href="arytm_3.html#D2">ARYTM_3:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E27:19"/><i><font color="Green">E137</font></i>: 
not <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"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="numbers.html#E7:19"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_5"/>
<font color="Maroon">two</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:19_5"/><b>then </b>
 ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Maroon">two</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font> )
 ;<br/><b>hence </b><a NAME="E3:19_5"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E8:19"><i><font color="Green">E29</font></i></a>, , , </i>;<br/></div><b>end;</b></div>
<a NAME="E29:19"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K7">RAT+</a> 
 
;<br/>
<a NAME="E30:19"/><b>then </b><i><font color="Green">E138</font></i>: 
not <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="arytm_3.html#K7">RAT+</a> </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><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> ;<br/><b>assume </b><a NAME="E1:19_6"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E3:19_6"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:19_6"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font>
 ;<br/><b>thus </b><a NAME="E5:19_6"/>
for <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">t</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r</font> holds <br/><font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="numbers.html#E13:19"><i><font color="Green">E35</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_6_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_6_1_1"><b>suppose </b></a><a NAME="E1:19_6_1_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>take </b><font color="Maroon">1'</font> = <font color="Maroon">1'</font>;<br/><b>thus </b><a NAME="E2:19_6_1_1"/>
<font color="Maroon">1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="numbers.html#E9"><i><font color="Green">Lemma22</font></i></a></i>;<br/><b>thus </b><a NAME="E3:19_6_1_1"/>
<font color="Maroon">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">1'</font>
 <b>by </b><i>, <a class="txt" href="numbers.html#E16:19"><i><font color="Green">E38</font></i></a>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_6_1_2"><b>suppose </b></a><a NAME="E1:19_6_1_2"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">rr</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E3:19_6_1_2"/><i><font color="Green">E145</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">rr</font> <a href="hidden.html#R1">=</a> <font color="Maroon">two</font> or <font color="Maroon">two</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">rr</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font> )
 <b>by </b><i><a class="ref" href="arytm_3.html#T100">ARYTM_3:100</a></i>;<br/><a NAME="E4:19_6_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="numbers.html#E16:19"><i><font color="Green">E38</font></i></a>, <a class="ref" href="arytm_3.html#T70">ARYTM_3:70</a></i>;<br/><b>then consider </b><font color="Maroon">3r'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E6:19_6_1_2"/><i><font color="Green">E147</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T60">ARYTM_3:60</a></i>;<br/><b>set </b><font color="Maroon">eps</font> = <font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font>;<br/><div><i><font color="Green">E149</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_6_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_6_1_2_1"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_6_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_6_1_2_1_1"/>
( <font color="Maroon">rr</font> <a href="hidden.html#R1">=</a> <font color="Maroon">0'</font> or <font color="Maroon">3r'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">0'</font> )
 </span><b>by </b><i>, <a class="ref" href="arytm_3.html#T86">ARYTM_3:86</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_6_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_6_1_2_1_1_1"/>
<font color="Maroon">rr</font> <a href="hidden.html#R1">=</a> <font color="Maroon">0'</font>
 ;<br/><div class="add"><a NAME="E2:19_6_1_2_1_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">two</font>
 <b>by </b><i><a class="txt" href="numbers.html#E17:19"><i><font color="Green">E39</font></i></a>, <a class="ref" href="arytm_3.html#T56">ARYTM_3:56</a></i>;<br/><b>hence </b><a NAME="E3:19_6_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E14:19"><i><font color="Green">E36</font></i></a>, <a class="txt" href="numbers.html#E15:19"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_6_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_6_1_2_1_1_2"/>
<font color="Maroon">3r'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">0'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_6_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E19:19"><i><font color="Green">E40</font></i></a>, <a class="ref" href="arytm_3.html#T54">ARYTM_3:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_6_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_6_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_6_1_2_2_1"/>
( <font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font> or <font color="Maroon">r</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_6_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:19_6_1_2_2_1_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">t</font> = <font color="Maroon">r</font> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span>;<br/><i><font color="Green">E152</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <span class="p3">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a></i>
<br/>.= 
<font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> 
<b>by </b><i><a class="txt" href="numbers.html#E19:19"><i><font color="Green">E40</font></i></a>, <a class="ref" href="arytm_3.html#T58">ARYTM_3:58</a></i>
<br/>.= 
<font color="Maroon">rr</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>
;<br/><a NAME="E3:19_6_1_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span>
 <b>by </b><i><a class="txt" href="numbers.html#E20:19"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="arytm_3.html#T88">ARYTM_3:88</a></i>;<br/><a NAME="E4:19_6_1_2_2_1_1"/><b>then </b><i><font color="Green">E153</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <span class="p3">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span></span>)</span> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <span class="p3">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="arytm_3.html#T83">ARYTM_3:83</a></i>;<br/><font color="Maroon">t</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t</font> = 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <span class="p3">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <span class="p3">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p3">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <span class="p3">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p4">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T57">ARYTM_3:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <span class="p4">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p3">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T57">ARYTM_3:57</a></i>
;<br/><a NAME="E6:19_6_1_2_2_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font>
 <b>by </b><i><a class="txt" href="numbers.html#E14:19"><i><font color="Green">E36</font></i></a>, <a class="txt" href="numbers.html#E15:19"><i><font color="Green">E37</font></i></a>, <a class="txt" href="numbers.html#E17:19"><i><font color="Green">E39</font></i></a>, , , <a class="ref" href="arytm_3.html#T83">ARYTM_3:83</a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/><b>hence </b><a NAME="E7:19_6_1_2_2_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E8:19_6_1_2_2_1_1"/>
<font color="Maroon">0'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T71">ARYTM_3:71</a></i>;<br/><a NAME="E9:19_6_1_2_2_1_1"/><b>then </b>
<font color="Maroon">0'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font>
 <b>by </b><i><a class="txt" href="numbers.html#E20:19"><i><font color="Green">E41</font></i></a>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/><a NAME="E10:19_6_1_2_2_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">0'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span>
 <b>by </b><i><a class="ref" href="arytm_3.html#T83">ARYTM_3:83</a></i>;<br/><b>hence </b><a NAME="E11:19_6_1_2_2_1_1"/>
<font color="Maroon">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T56">ARYTM_3:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_6_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:19_6_1_2_2_1_2"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">r</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">t</font> = <span class="p1">(<span class="default"><font color="Maroon">1'</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">half</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font>;<br/><span class="p1">(<span class="default"><font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">3r'</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r</font></span>)</span> = 
<font color="Maroon">rr</font> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> 
<b>by </b><i><a class="txt" href="numbers.html#E19:19"><i><font color="Green">E40</font></i></a>, <a class="ref" href="arytm_3.html#T58">ARYTM_3:58</a></i>
<br/>.= 
<font color="Maroon">rr</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>
;<br/><a NAME="E3:19_6_1_2_2_1_2"/><b>then </b><i><font color="Green">E155</font></i>: 
<font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">rr</font>
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T90">ARYTM_3:90</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T57">ARYTM_3:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T58">ARYTM_3:58</a></i>
<br/>.= 
<font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T58">ARYTM_3:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T58">ARYTM_3:58</a></i>
;<br/><a NAME="E5:19_6_1_2_2_1_2"/><b>then </b><i><font color="Green">E156</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">two</font>
 <b>by </b><i><a class="txt" href="numbers.html#E14:19"><i><font color="Green">E36</font></i></a>, <a class="txt" href="numbers.html#E15:19"><i><font color="Green">E37</font></i></a>, <a class="txt" href="numbers.html#E17:19"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="arytm_3.html#T83">ARYTM_3:83</a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/><a NAME="E6:19_6_1_2_2_1_2"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="numbers.html#E16:19"><i><font color="Green">E38</font></i></a>, <a class="ref" href="arytm_3.html#T86">ARYTM_3:86</a></i>;<br/><a NAME="E7:19_6_1_2_2_1_2"/>
<font color="Maroon">1'</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">half</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="arytm_3.html#T70">ARYTM_3:70</a></i>;<br/><a NAME="E8:19_6_1_2_2_1_2"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="numbers.html#E16:19"><i><font color="Green">E38</font></i></a>, <a class="ref" href="arytm_3.html#T86">ARYTM_3:86</a></i>;<br/><a NAME="E9:19_6_1_2_2_1_2"/>
<font color="Maroon">1'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i><a class="txt" href="numbers.html#E9"><i><font color="Green">Lemma22</font></i></a>, <a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>;<br/><a NAME="E10:19_6_1_2_2_1_2"/><b>then </b>
<font color="Maroon">half</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i><a class="txt" href="numbers.html#E8"><i><font color="Green">Lemma16</font></i></a>, <a class="ref" href="arytm_3.html#T90">ARYTM_3:90</a></i>;<br/><a NAME="E11:19_6_1_2_2_1_2"/><b>then </b>
<a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K11">+</a> <font color="Maroon">half</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font>
 <b>by </b><i><a class="txt" href="numbers.html#E5:19"><i><font color="Green">E13</font></i></a>, <a class="ref" href="arytm_3.html#T83">ARYTM_3:83</a></i>;<br/><a NAME="E12:19_6_1_2_2_1_2"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">t</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="numbers.html#E16:19"><i><font color="Green">E38</font></i></a>, <a class="ref" href="arytm_3.html#T88">ARYTM_3:88</a></i>;<br/><a NAME="E13:19_6_1_2_2_1_2"/><b>then </b><i><font color="Green">E160</font></i>: 
<font color="Maroon">t</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="numbers.html#E21:19"><i><font color="Green">E46</font></i></a>, <a class="ref" href="arytm_3.html#T88">ARYTM_3:88</a></i>;<br/><a NAME="E14:19_6_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="numbers.html#E22:19"><i><font color="Green">E47</font></i></a>, <a class="ref" href="arytm_3.html#T88">ARYTM_3:88</a></i>;<br/><a NAME="E15:19_6_1_2_2_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><font color="Maroon">two</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="numbers.html#E25:19"><i><font color="Green">E48</font></i></a>, <a class="ref" href="arytm_3.html#T77">ARYTM_3:77</a></i>;<br/><a NAME="E16:19_6_1_2_2_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font>
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T76">ARYTM_3:76</a></i>;<br/><b>hence </b><a NAME="E17:19_6_1_2_2_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E18:19_6_1_2_2_1_2"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">0'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">half</font>
 <b>by </b><i><a class="txt" href="numbers.html#E8"><i><font color="Green">Lemma16</font></i></a>, <a class="ref" href="arytm_3.html#T54">ARYTM_3:54</a></i>;<br/><a NAME="E19:19_6_1_2_2_1_2"/>
<font color="Maroon">0'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">half</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T71">ARYTM_3:71</a></i>;<br/><a NAME="E20:19_6_1_2_2_1_2"/><b>then </b>
<font color="Maroon">0'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">half</font>
 <b>by </b><i><a class="txt" href="numbers.html#E26:19"><i><font color="Green">E49</font></i></a>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/><a NAME="E21:19_6_1_2_2_1_2"/><b>then </b>
<a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K11">+</a> <font color="Maroon">0'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K11">+</a> <font color="Maroon">half</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T83">ARYTM_3:83</a></i>;<br/><a NAME="E22:19_6_1_2_2_1_2"/><b>then </b>
 <a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#NR4" target="_self">&lt;</a> <a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K11">+</a> <font color="Maroon">half</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T56">ARYTM_3:56</a></i>;<br/><a NAME="E23:19_6_1_2_2_1_2"/><b>then </b>
<a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K12">*'</a> <font color="Maroon">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="numbers.html#E16:19"><i><font color="Green">E38</font></i></a>, <a class="ref" href="arytm_3.html#T88">ARYTM_3:88</a></i>;<br/><b>hence </b><a NAME="E24:19_6_1_2_2_1_2"/>
<font color="Maroon">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:19_6_1_2"/>
 ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">t</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E32:19"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/>( ( for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">r</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) &amp;  ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">s</font> ) ) </span>}</span> 
 
;<br/>
<a NAME="E33:19"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K1">DEDEKIND_CUTS</a> 
 
<b>by </b><i><a class="txt" href="numbers.html#E12:19"><i><font color="Green">E34</font></i></a>, <a class="ref" href="arytm_2.html#D1">ARYTM_2:def 1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E34:19"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <a href="arytm_3.html#K7">RAT+</a>  <a href="xboole_0.html#K2">\/</a> <a href="arytm_2.html#K1">DEDEKIND_CUTS</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E35:19"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 
<b>by </b><i>, <a class="ref" href="arytm_2.html#D2">ARYTM_2:def 2</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E36:19"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <a href="arytm_2.html#K2">REAL+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E37:19"/><b>then </b><i><font color="Green">E162</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1" target="_self">REAL</a> 
 
<b>by </b><i><a class="txt" href="numbers.html#E9:19"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_7"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3" target="_self">RAT</a> 
 ;<br/><a NAME="E2:19_7"/><b>then </b><i><font color="Green">E163</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <a href="arytm_3.html#K7">RAT+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_3.html#K7">RAT+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_7_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  or <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_3.html#K7">RAT+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 </span><b>by </b><i><a class="txt" href="numbers.html#E14:19"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_7_1_1"><b>suppose </b></a><a NAME="E1:19_7_1_1"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_7_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_7_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_7_1_1_1_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a>  : ( <font color="Olive">i</font>,<font color="Olive">j</font> <a href="arytm_3.html#R1">are_relative_prime</a>  &amp; <font color="Olive">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  ) </span>}</span>  <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<a href="arytm_3.html#K14">one</a> </span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a>  : verum </span>}</span>  or <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  )
 </span><b>by </b><i><a class="txt" href="numbers.html#E15:19"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_7_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:19_7_1_1_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a>  : ( <font color="Olive">i</font>,<font color="Olive">j</font> <a href="arytm_3.html#R1">are_relative_prime</a>  &amp; <font color="Olive">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  ) </span>}</span>  <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<a href="arytm_3.html#K14">one</a> </span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a>  : verum </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:19_7_1_1_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a>  : ( <font color="Olive">i</font>,<font color="Olive">j</font> <a href="arytm_3.html#R1">are_relative_prime</a>  &amp; <font color="Olive">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a> <b> such that </b><br/><a NAME="E4:19_7_1_1_1_1_1"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E5:19_7_1_1_1_1_1"/>
( <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="arytm_3.html#R1">are_relative_prime</a>  &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  )
 ;<br/><a NAME="E6:19_7_1_1_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="numbers.html#E16:19"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D5">TARSKI:def 5</a></i>;<br/><b>hence </b><a NAME="E7:19_7_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E6:19"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_7_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:19_7_1_1_1_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/><div class="add"><a NAME="E2:19_7_1_1_1_1_2"/><b>then </b><i><font color="Green">E166</font></i>: 
<font color="Maroon">X</font> is  <a href="ordinal1.html#V3">ordinal</a>  <a href="ordinal1.html#NM1" target="_self">number</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><a NAME="E3:19_7_1_1_1_1_2"/>
2 <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="numbers.html#E6:19"><i><font color="Green">E25</font></i></a>, <a class="txt" href="numbers.html#E1"><i><font color="Green">Lemma11</font></i></a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/><a NAME="E4:19_7_1_1_1_1_2"/><b>then </b><i><font color="Green">E167</font></i>: 
not <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> 2
 <b>by </b><i><a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_7_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_7_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_7_1_1_1_1_2_1_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">two</font> or <font color="Maroon">two</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 </span><b>by </b><i><a class="txt" href="numbers.html#E16:19"><i><font color="Green">E38</font></i></a>, <a class="txt" href="numbers.html#E17:19"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal1.html#T24">ORDINAL1:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_7_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_7_1_1_1_1_2_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">two</font>
 ;<br/><div class="add"><a NAME="E2:19_7_1_1_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">half</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">half</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, <a class="txt" href="numbers.html#E1"><i><font color="Green">Lemma11</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:19_7_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E8"><i><font color="Green">Lemma16</font></i></a>, <a class="ref" href="arytm_3.html#T54">ARYTM_3:54</a>, <a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_7_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_7_1_1_1_1_2_1_1_2"/>
<font color="Maroon">two</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:19_7_1_1_1_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">two</font> &amp; <font color="Olive">s</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">two</font> )
 ;<br/><b>hence </b><a NAME="E3:19_7_1_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E8:19"><i><font color="Green">E29</font></i></a>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:19_7_1_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:19_7_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_7_1_2"><b>suppose </b></a><a NAME="E1:19_7_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_3.html#K7">RAT+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_7_1_2"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/><a NAME="E4:19_7_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="numbers.html#E15:19"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D5">TARSKI:def 5</a></i>;<br/><b>hence </b><a NAME="E5:19_7_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E6:19"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E39:19"/>
 <a href="numbers.html#K3" target="_self">RAT</a>  <a href="xboole_0.html#R2">c&lt;</a>  <a href="numbers.html#K1" target="_self">REAL</a> 
 <b>by </b><i><a class="txt" href="numbers.html#E13:19"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>


</div>
