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

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> ] <span class="kw">means </span>$1 <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> $1 <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>] </span>}</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>] </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  <span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></span>
<a NAME="E2:20"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
( 2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c2">two</font> &amp; 1 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T65" title="ARYTM_3:th.65">ARYTM_3:65</a>, <a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a></span>;<br/>
2 = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> 1

<br/>.= 
1 <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D1" title="ORDINAL1:def.1">ORDINAL1:def 1</a></span>
;<br/>
<a NAME="E4:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c1">a9</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c2">two</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="numbers.html#E15"><span class="lab"><font color="Green" title="E11">Lm10</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E5:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c1">a9</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>;<br/>
<a NAME="E6:20"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c1">a9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>;<br/>
<a NAME="E7:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A5</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E19">A3</font></span></a></span>;<br/>
<a NAME="E8:20"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Olive" title="b2">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c6">t</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c6">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r</font> implies <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:20_2"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r</font> or <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )</span><br/>

<a NAME="E2:20_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A7</font></span>: 
 ex <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font> )
 
;<br/>
<span class="kw">assume </span><a NAME="E3:20_2"/>
<font color="Maroon" title="c6">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font></span><br/>

<a NAME="E4:20_2"/><span class="kw">then </span>
( <font color="Maroon" title="c6">t</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c6">t</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon" title="c6">t</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></span>;<br/>
<a NAME="E5:20_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">t</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T74" title="ARYTM_3:th.74">ARYTM_3:74</a></span>;<br/>
<a NAME="E6:20_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">t</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="arytm_3.html#T76" title="ARYTM_3:th.76">ARYTM_3:76</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:20_2"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A8</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="arytm_3.html#T71" title="ARYTM_3:th.71">ARYTM_3:71</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_3"/>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><font color="Maroon" title="c4">X</font> = 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:20_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E11:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A9</font></span>: 
not <font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">09</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="numbers.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">DD</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>   st <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/>( ( for <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>   st <font color="Olive" title="b3">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b2">r</font> holds <br/><font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> ) &amp;  ex <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b3">s</font> ) ) </span>}</span> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">half</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <span class="kw"> such that </span><br/><a NAME="E14:20"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
<font color="Maroon" title="c1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T61" title="ARYTM_3:th.61">ARYTM_3:61</a></span>;<br/>
<a NAME="E15:20"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
 <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c2">two</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="numbers.html#E22"><span class="lab"><font color="Green" title="E14">Lm13</font></span></a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>;<br/>
<a NAME="E16:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A12</font></span>: 
 <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E28">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_4"/>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive" title="b2">s</font> where <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : <font color="Olive" title="b2">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">t</font> </span>}</span>  where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : <font color="Olive" title="b1">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">t0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <span class="kw"> such that </span><br/><a NAME="E3:20_4"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">t0</font> </span>}</span> 
 <span class="kw">and </span><br/><a NAME="E4:20_4"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<font color="Maroon" title="c8">t0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><span class="kw">set </span><font color="Maroon" title="c9">n</font> =  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c8">t0</font>;<br/><span class="kw">set </span><font color="Maroon" title="c10">d</font> =  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c8">t0</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_4_1"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
<font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">two</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_4_1_1"/>
( <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font> or <font color="Maroon" title="c2">two</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_4_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="arytm_3.html#T73" title="ARYTM_3:th.73">ARYTM_3:73</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_4_1_1_1"/>
<font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:20_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/><a NAME="E3:20_4_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t0</font> &amp; <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">t0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_4"><span class="lab"><font color="Green" title="E29">A14</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:20_4_1_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_4_1_1_2"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
<font color="Maroon" title="c2">two</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c11">es</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <span class="kw"> such that </span><br/><a NAME="E3:20_4_1_1_2"/><span class="lab"><font color="Green" title="E33">A18</font></span>: 
( <font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c11">es</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c11">es</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">two</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T100" title="ARYTM_3:th.100">ARYTM_3:100</a></span>;<br/><div><span class="lab"><font color="Green" title="E34">A19</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_4_1_1_2_1"/>
<font color="Maroon" title="c5">09</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">es</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:20_4_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c11">es</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">two</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T56" title="ARYTM_3:th.56">ARYTM_3:56</a></span>;<br/><span class="kw">hence </span><a NAME="E3:20_4_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_4_1_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E3:20_4_1_1_2"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E5:20_4_1_1_2"/>
<font color="Maroon" title="c5">09</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c11">es</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T71" title="ARYTM_3:th.71">ARYTM_3:71</a></span>;<br/><a NAME="E6:20_4_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">09</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c11">es</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_4_1_1_2"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <span class="kw"> such that </span><br/><a NAME="E8:20_4_1_1_2"/><span class="lab"><font color="Green" title="E35">A20</font></span>: 
<font color="Maroon" title="c5">09</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c12">s</font>
 <span class="kw">and </span><br/><a NAME="E9:20_4_1_1_2"/><span class="lab"><font color="Green" title="E36">A21</font></span>: 
<font color="Maroon" title="c12">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c11">es</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T101" title="ARYTM_3:th.101">ARYTM_3:101</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1_1_2_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_4_1_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_4_1_1_2_2_1"/>
( <font color="Maroon" title="c12">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  or  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c12">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_4_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_4_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E37">A22</font></span>: 
<font color="Maroon" title="c12">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:20_4_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E38">A23</font></span>: 
<font color="Maroon" title="c12">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20_4_1_1_2"><span class="lab"><font color="Green" title="E35">A20</font></span></a></span>;<br/><a NAME="E3:20_4_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="arytm_3.html#T88" title="ARYTM_3:th.88">ARYTM_3:88</a></span>;<br/><a NAME="E4:20_4_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A24</font></span>: 
<font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c12">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>;<br/><div><span class="lab"><font color="Green" title="E40">A25</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1_1_2_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_4_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E41">A26</font></span>: 
<font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:20_4_1_1_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></span>;<br/><a NAME="E3:20_4_1_1_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>;<br/><a NAME="E4:20_4_1_1_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_4_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="ref" href="arytm_3.html#T74" title="ARYTM_3:th.74">ARYTM_3:74</a></span>;<br/><span class="kw">hence </span><a NAME="E5:20_4_1_1_2_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:20"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E1:20_4_1_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="arytm_3.html#T76" title="ARYTM_3:th.76">ARYTM_3:76</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:20_4_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A27</font></span>: 
<a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T88" title="ARYTM_3:th.88">ARYTM_3:88</a></span>;<br/><a NAME="E7:20_4_1_1_2_2_1_1"/>
<a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:20"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E4:20_4"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="arytm_3.html#T88" title="ARYTM_3:th.88">ARYTM_3:88</a></span>;<br/><a NAME="E8:20_4_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A28</font></span>: 
<a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="ref" href="arytm_3.html#T77" title="ARYTM_3:th.77">ARYTM_3:77</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">t02s2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <span class="kw"> such that </span><br/><a NAME="E10:20_4_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E43">A29</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c13">t02s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c13">t02s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T100" title="ARYTM_3:th.100">ARYTM_3:100</a></span>;<br/><a NAME="E11:20_4_1_1_2_2_1_1"/>
<font color="Maroon" title="c12">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E5:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="ref" href="arytm_3.html#T77" title="ARYTM_3:th.77">ARYTM_3:77</a></span>;<br/><a NAME="E12:20_4_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A30</font></span>: 
<font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="ref" href="arytm_3.html#T77" title="ARYTM_3:th.77">ARYTM_3:77</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">2t9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <span class="kw"> such that </span><br/><a NAME="E14:20_4_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E45">A31</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_4"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="arytm_3.html#T61" title="ARYTM_3:th.61">ARYTM_3:61</a>, <a class="ref" href="arytm_3.html#T86" title="ARYTM_3:th.86">ARYTM_3:86</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c15">x</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">t0x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <span class="kw"> such that </span><br/><a NAME="E16:20_4_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E46">A32</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c16">t0x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t0</font> or <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c16">t0x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T100" title="ARYTM_3:th.100">ARYTM_3:100</a></span>;<br/><a NAME="E17:20_4_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="ref" href="arytm_3.html#T58" title="ARYTM_3:th.58">ARYTM_3:58</a></span>;<br/><a NAME="E18:20_4_1_1_2_2_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font> or <font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T91" title="ARYTM_3:th.91">ARYTM_3:91</a></span>;<br/><a NAME="E19:20_4_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E12:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a>, <a class="ref" href="arytm_3.html#T76" title="ARYTM_3:th.76">ARYTM_3:76</a></span>;<br/><a NAME="E20:20_4_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A34</font></span>: 
<font color="Maroon" title="c16">t0x</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>;<br/><span class="lab"><font color="Green" title="E49">A35</font></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" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c16">t0x</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</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" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c16">t0x</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="txt" href="#E19:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>
<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" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> </span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>
<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" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> </span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</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" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c2">two</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="numbers.html#E22"><span class="lab"><font color="Green" title="E14">Lm13</font></span></a>, <a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c2">two</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T58" title="ARYTM_3:th.58">ARYTM_3:58</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="ref" href="arytm_3.html#T58" title="ARYTM_3:th.58">ARYTM_3:58</a></span>
<br/>.= 
<font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>
;<br/><a NAME="E22:20_4_1_1_2_2_1_1"/>
<font color="Maroon" title="c11">es</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_4_1_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E3:20_4_1_1_2"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>;<br/><a NAME="E23:20_4_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20_4_1_1_2"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="arytm_3.html#T76" title="ARYTM_3:th.76">ARYTM_3:76</a></span>;<br/><a NAME="E24:20_4_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A36</font></span>: 
<font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="ref" href="arytm_3.html#T77" title="ARYTM_3:th.77">ARYTM_3:77</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c13">t02s2</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">t0x</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E16:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="txt" href="#E19:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">t0x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">t0x</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="txt" href="#E19:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">t0x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c16">t0x</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c16">t0x</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">t0x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c16">t0x</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c16">t0x</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</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" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">t0x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c16">t0x</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</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" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c16">t0x</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</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" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">t0x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c16">t0x</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E49">A35</font></span></a>, <a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a></span>
;<br/><a NAME="E26:20_4_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">t0x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c16">t0x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t02s2</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T69" title="ARYTM_3:th.69">ARYTM_3:69</a></span>;<br/><a NAME="E27:20_4_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A37</font></span>: 
<font color="Maroon" title="c13">t02s2</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c16">t0x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c16">t0x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1_1_2_2_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_4_1_1_2_2_1_1_4"/><span class="lab"><font color="Green" title="E52">A38</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">2t9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_4_1_1_2_2_1_1_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_4_1_1_2_2_1_1_4_1"/>
( <font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c14">2t9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_4_1_1_2_2_1_1_4"><span class="lab"><font color="Green" title="E52">A38</font></span></a>, <a class="ref" href="arytm_3.html#T86" title="ARYTM_3:th.86">ARYTM_3:86</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_4_1_1_2_2_1_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_4_1_1_2_2_1_1_4_1_1"/>
<font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_4_1_1_2_2_1_1_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="arytm_3.html#T86" title="ARYTM_3:th.86">ARYTM_3:86</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_4_1_1_2_2_1_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_4_1_1_2_2_1_1_4_1_2"/>
<font color="Maroon" title="c14">2t9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_4_1_1_2_2_1_1_4_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E29:20_4_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">t0x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="txt" href="#E19:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="ref" href="arytm_3.html#T92" title="ARYTM_3:th.92">ARYTM_3:92</a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>;<br/><a NAME="E30:20_4_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">t0x</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>;<br/><a NAME="E31:20_4_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">t0x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_4"><span class="lab"><font color="Green" title="E29">A14</font></span></a></span>;<br/><a NAME="E32:20_4_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A39</font></span>: 
 ex <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">t0x</font> &amp; <font color="Olive" title="b1">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font> )
 ;<br/><a NAME="E33:20_4_1_1_2_2_1_1"/>
<font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c11">es</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20_4_1_1_2"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E4:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="ref" href="arytm_3.html#T77" title="ARYTM_3:th.77">ARYTM_3:77</a></span>;<br/><a NAME="E34:20_4_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c11">es</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T83" title="ARYTM_3:th.83">ARYTM_3:83</a></span>;<br/><a NAME="E35:20_4_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">two</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c13">t02s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_4_1_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E3:20_4_1_1_2"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E10:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E24:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="ref" href="arytm_3.html#T83" title="ARYTM_3:th.83">ARYTM_3:83</a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E36:20_4_1_1_2_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E51">A37</font></span></a>, <a class="txt" href="#E32:20_4_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="ref" href="arytm_3.html#T76" title="ARYTM_3:th.76">ARYTM_3:76</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_4_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_4_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E37">A40</font></span>: 
 <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c12">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:20_4_1_1_2_2_1_2"/>
<font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c2">two</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>;<br/><a NAME="E3:20_4_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A41</font></span>: 
<font color="Maroon" title="c7">half</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:20"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="arytm_3.html#T91" title="ARYTM_3:th.91">ARYTM_3:91</a></span>;<br/><a NAME="E4:20_4_1_1_2_2_1_2"/>
<font color="Maroon" title="c7">half</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>;<br/><a NAME="E5:20_4_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A42</font></span>: 
<font color="Maroon" title="c7">half</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E38">A41</font></span></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>;<br/><a NAME="E6:20_4_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">half</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c12">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E37">A40</font></span></a>, <a class="ref" href="arytm_3.html#T76" title="ARYTM_3:th.76">ARYTM_3:76</a></span>;<br/><a NAME="E7:20_4_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A43</font></span>: 
<font color="Maroon" title="c7">half</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c11">es</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20_4_1_1_2"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="arytm_3.html#T77" title="ARYTM_3:th.77">ARYTM_3:77</a></span>;<br/><a NAME="E8:20_4_1_1_2_2_1_2"/>
 <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c2">two</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="numbers.html#E22"><span class="lab"><font color="Green" title="E14">Lm13</font></span></a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>;<br/><a NAME="E9:20_4_1_1_2_2_1_2"/><span class="kw">then </span>
 <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>;<br/><a NAME="E10:20_4_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A44</font></span>: 
<a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_4"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="arytm_3.html#T88" title="ARYTM_3:th.88">ARYTM_3:88</a></span>;<br/><div><span class="lab"><font color="Green" title="E42">A45</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1_1_2_2_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_4_1_1_2_2_1_2_1"/><span class="lab"><font color="Green" title="E43">A46</font></span>: 
<font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:20_4_1_1_2_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></span>;<br/><a NAME="E3:20_4_1_1_2_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>;<br/><a NAME="E4:20_4_1_1_2_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_4_1_1_2_2_1_2_1"><span class="lab"><font color="Green" title="E43">A46</font></span></a>, <a class="ref" href="arytm_3.html#T74" title="ARYTM_3:th.74">ARYTM_3:74</a></span>;<br/><span class="kw">hence </span><a NAME="E5:20_4_1_1_2_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:20"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E1:20_4_1_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="arytm_3.html#T76" title="ARYTM_3:th.76">ARYTM_3:76</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E12:20_4_1_1_2_2_1_2"/><span class="kw">then </span>
<a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T88" title="ARYTM_3:th.88">ARYTM_3:88</a></span>;<br/><a NAME="E13:20_4_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A47</font></span>: 
<a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E41">A44</font></span></a>, <a class="ref" href="arytm_3.html#T77" title="ARYTM_3:th.77">ARYTM_3:77</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">s</font> = <font color="Maroon" title="c7">half</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">t02s2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <span class="kw"> such that </span><br/><a NAME="E15:20_4_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E44">A48</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c14">t02s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c14">t02s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T100" title="ARYTM_3:th.100">ARYTM_3:100</a></span>;<br/><a NAME="E16:20_4_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E45">A49</font></span>: 
<font color="Maroon" title="c7">half</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></span>;<br/><a NAME="E17:20_4_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="ref" href="arytm_3.html#T88" title="ARYTM_3:th.88">ARYTM_3:88</a></span>;<br/><a NAME="E18:20_4_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A50</font></span>: 
<font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c7">half</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>;<br/><a NAME="E19:20_4_1_1_2_2_1_2"/>
<font color="Maroon" title="c7">half</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="txt" href="#E11:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E42">A45</font></span></a>, <a class="ref" href="arytm_3.html#T77" title="ARYTM_3:th.77">ARYTM_3:77</a></span>;<br/><a NAME="E20:20_4_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A51</font></span>: 
<font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E46">A50</font></span></a>, <a class="ref" href="arytm_3.html#T77" title="ARYTM_3:th.77">ARYTM_3:77</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">2t9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <span class="kw"> such that </span><br/><a NAME="E22:20_4_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E48">A52</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_4"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="arytm_3.html#T61" title="ARYTM_3:th.61">ARYTM_3:61</a>, <a class="ref" href="arytm_3.html#T86" title="ARYTM_3:th.86">ARYTM_3:86</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c16">x</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">t0x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <span class="kw"> such that </span><br/><a NAME="E24:20_4_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E49">A53</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c17">t0x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t0</font> or <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c17">t0x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T100" title="ARYTM_3:th.100">ARYTM_3:100</a></span>;<br/><a NAME="E25:20_4_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E48">A52</font></span></a>, <a class="ref" href="arytm_3.html#T58" title="ARYTM_3:th.58">ARYTM_3:58</a></span>;<br/><a NAME="E26:20_4_1_1_2_2_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font> or <font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T91" title="ARYTM_3:th.91">ARYTM_3:91</a></span>;<br/><a NAME="E27:20_4_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A54</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E43">A47</font></span></a>, <a class="txt" href="#E20:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E47">A51</font></span></a>, <a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a>, <a class="ref" href="arytm_3.html#T76" title="ARYTM_3:th.76">ARYTM_3:76</a></span>;<br/><a NAME="E28:20_4_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A55</font></span>: 
<font color="Maroon" title="c17">t0x</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E49">A53</font></span></a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>;<br/><span class="lab"><font color="Green" title="E52">A56</font></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" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c17">t0x</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</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" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c17">t0x</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E49">A53</font></span></a>, <a class="txt" href="#E27:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E50">A54</font></span></a>, <a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>
<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" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> </span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>
<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" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> </span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</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" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c2">two</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="numbers.html#E22"><span class="lab"><font color="Green" title="E14">Lm13</font></span></a>, <a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c2">two</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T58" title="ARYTM_3:th.58">ARYTM_3:58</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E48">A52</font></span></a>, <a class="ref" href="arytm_3.html#T58" title="ARYTM_3:th.58">ARYTM_3:58</a></span>
<br/>.= 
<font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>
;<br/><a NAME="E30:20_4_1_1_2_2_1_2"/>
<font color="Maroon" title="c11">es</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_4_1_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E3:20_4_1_1_2"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>;<br/><a NAME="E31:20_4_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">half</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E40">A43</font></span></a>, <a class="ref" href="arytm_3.html#T76" title="ARYTM_3:th.76">ARYTM_3:76</a></span>;<br/><a NAME="E32:20_4_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A57</font></span>: 
<font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E46">A50</font></span></a>, <a class="ref" href="arytm_3.html#T77" title="ARYTM_3:th.77">ARYTM_3:77</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c14">t02s2</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E44">A48</font></span></a>, <a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">t0x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">t0x</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E49">A53</font></span></a>, <a class="txt" href="#E27:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E50">A54</font></span></a>, <a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">t0x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c17">t0x</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c17">t0x</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">t0x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c17">t0x</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c17">t0x</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</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" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">t0x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c17">t0x</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</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" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c17">t0x</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</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" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">t0x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c17">t0x</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A56</font></span></a>, <a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a></span>
;<br/><a NAME="E34:20_4_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">t0x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c17">t0x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">t02s2</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T69" title="ARYTM_3:th.69">ARYTM_3:69</a></span>;<br/><a NAME="E35:20_4_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A58</font></span>: 
<font color="Maroon" title="c14">t02s2</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c17">t0x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c17">t0x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1_1_2_2_1_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_4_1_1_2_2_1_2_4"/><span class="lab"><font color="Green" title="E55">A59</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c15">2t9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_4_1_1_2_2_1_2_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_4_1_1_2_2_1_2_4_1"/>
( <font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c15">2t9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_4_1_1_2_2_1_2_4"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="ref" href="arytm_3.html#T86" title="ARYTM_3:th.86">ARYTM_3:86</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_4_1_1_2_2_1_2_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_4_1_1_2_2_1_2_4_1_1"/>
<font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_4_1_1_2_2_1_2_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E45">A49</font></span></a>, <a class="ref" href="arytm_3.html#T86" title="ARYTM_3:th.86">ARYTM_3:86</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_4_1_1_2_2_1_2_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_4_1_1_2_2_1_2_4_1_2"/>
<font color="Maroon" title="c15">2t9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_4_1_1_2_2_1_2_4_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E48">A52</font></span></a>, <a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E37:20_4_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">t0x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E49">A53</font></span></a>, <a class="txt" href="#E27:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E50">A54</font></span></a>, <a class="ref" href="arytm_3.html#T92" title="ARYTM_3:th.92">ARYTM_3:92</a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>;<br/><a NAME="E38:20_4_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">t0x</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E51">A55</font></span></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>;<br/><a NAME="E39:20_4_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">t0x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_4"><span class="lab"><font color="Green" title="E29">A14</font></span></a></span>;<br/><a NAME="E40:20_4_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A60</font></span>: 
 ex <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t0x</font> &amp; <font color="Olive" title="b1">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font> )
 ;<br/><a NAME="E41:20_4_1_1_2_2_1_2"/>
<font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c11">es</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E46">A50</font></span></a>, <a class="txt" href="#E7:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E40">A43</font></span></a>, <a class="ref" href="arytm_3.html#T77" title="ARYTM_3:th.77">ARYTM_3:77</a></span>;<br/><a NAME="E42:20_4_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">half</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c11">es</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T83" title="ARYTM_3:th.83">ARYTM_3:83</a></span>;<br/><a NAME="E43:20_4_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">two</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c14">t02s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_4_1_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E3:20_4_1_1_2"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E15:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E44">A48</font></span></a>, <a class="txt" href="#E32:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E53">A57</font></span></a>, <a class="ref" href="arytm_3.html#T83" title="ARYTM_3:th.83">ARYTM_3:83</a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E44:20_4_1_1_2_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E54">A58</font></span></a>, <a class="txt" href="#E40:20_4_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E55">A60</font></span></a>, <a class="ref" href="arytm_3.html#T76" title="ARYTM_3:th.76">ARYTM_3:76</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:20_4_1_1_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E6:20_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A61</font></span>: 
<font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c8">t0</font></span>)</span></span>)</span> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c8">t0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T46" title="ARYTM_3:th.46">ARYTM_3:46</a></span>;<br/><a NAME="E7:20_4"/>
 <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c8">t0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T41" title="ARYTM_3:th.41">ARYTM_3:41</a></span>;<br/><a NAME="E8:20_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T34" title="ORDINAL3:th.34">ORDINAL3:34</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A62</font></span>: <font color="Maroon" title="c2">two</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c8">t0</font></span>)</span></span>)</span> = 
1 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c8">t0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_4"><span class="lab"><font color="Green" title="E31">A61</font></span></a>, <a class="ref" href="arytm_3.html#T51" title="ARYTM_3:th.51">ARYTM_3:51</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c8">t0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a></span>
;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">k</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E11:20_4"/><span class="lab"><font color="Green" title="E33">A63</font></span>: 
 <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c8">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="numbers.html#E17"><span class="lab"><font color="Green" title="E13">Lm12</font></span></a></span>;<br/><a NAME="E12:20_4"/>
<font color="Maroon" title="c2">two</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c8">t0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default">2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c11">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20_4"><span class="lab"><font color="Green" title="E32">A62</font></span></a>, <a class="txt" href="#E11:20_4"><span class="lab"><font color="Green" title="E33">A63</font></span></a>, <a class="ref" href="ordinal3.html#T58" title="ORDINAL3:th.58">ORDINAL3:58</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c8">t0</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c8">t0</font></span>)</span> = 
<font color="Maroon" title="c11">k</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default">2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c11">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T36" title="ORDINAL3:th.36">ORDINAL3:36</a></span>
<br/>.= 
2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c11">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T58" title="ORDINAL3:th.58">ORDINAL3:58</a></span>
;<br/><a NAME="E14:20_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A64</font></span>: 
 ex <font color="Olive" title="b1">p</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c8">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="numbers.html#E17"><span class="lab"><font color="Green" title="E13">Lm12</font></span></a></span>;<br/><a NAME="E15:20_4"/>
 <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c8">t0</font>, <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T40" title="ARYTM_3:th.40">ARYTM_3:40</a></span>;<br/><span class="kw">hence </span><a NAME="E16:20_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:20_4"><span class="lab"><font color="Green" title="E33">A63</font></span></a>, <a class="txt" href="#E14:20_4"><span class="lab"><font color="Green" title="E34">A64</font></span></a>, <a class="ref" href="arytm_3.html#D2" title="ARYTM_3:def.2">ARYTM_3:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E18:20"/>
2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> 1
 
;<br/>
<a NAME="E19:20"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<a NAME="E20:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A65</font></span>: 
1 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T22" title="ORDINAL3:th.22">ORDINAL3:22</a></span>;<br/>
<a NAME="E21:20"/><span class="lab"><font color="Green" title="E30">A66</font></span>: 
<font color="Maroon" title="c5">09</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c1">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T71" title="ARYTM_3:th.71">ARYTM_3:71</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies ( ( for <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>   st <font color="Olive" title="b1">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">r</font> holds <br/><font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> ) &amp;  ex <font color="Olive" title="b1">a9</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Olive" title="b1">a9</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) ) )</span><br/><span class="kw">assume </span><a NAME="E1:20_5"/><span class="lab"><font color="Green" title="E31">A67</font></span>: 
<font color="Maroon" title="c8">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>   st <font color="Olive" title="b1">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">r</font> holds <br/><font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> ) &amp;  ex <font color="Olive" title="b1">a9</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Olive" title="b1">a9</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) )</span><br/><a NAME="E2:20_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A68</font></span>: 
 ex <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font> )
 ;<br/><span class="kw">thus </span><a NAME="E3:20_5"/>
for <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>   st <font color="Olive" title="b1">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">r</font> holds <br/><font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E1:20_5"><span class="lab"><font color="Green" title="E31">A67</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a9</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Olive" title="b1">a9</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_5_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_5_1"/>
( <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c8">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_5_1_1"/><span class="lab"><font color="Green" title="E33">A69</font></span>: 
<font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a9</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Olive" title="b1">a9</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c9">a9</font> = <font color="Maroon" title="c1">a9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c8">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c9">a9</font> )</span><br/><span class="kw">thus </span><a NAME="E2:20_5_1_1"/>
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E19">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c9">a9</font></span><br/><span class="kw">thus </span><a NAME="E3:20_5_1_1"/>
<font color="Maroon" title="c8">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c9">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E30">A66</font></span></a>, <a class="txt" href="#E1:20_5_1_1"><span class="lab"><font color="Green" title="E33">A69</font></span></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_5_1_2"/><span class="lab"><font color="Green" title="E33">A70</font></span>: 
<font color="Maroon" title="c8">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Olive" title="b1">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">3r9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <span class="kw"> such that </span><br/><a NAME="E3:20_5_1_2"/><span class="lab"><font color="Green" title="E34">A71</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T60" title="ARYTM_3:th.60">ARYTM_3:60</a>, <a class="ref" href="arytm_3.html#T70" title="ARYTM_3:th.70">ARYTM_3:70</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">rr</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <span class="kw"> such that </span><br/><a NAME="E5:20_5_1_2"/><span class="lab"><font color="Green" title="E35">A72</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c10">rr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">two</font> or <font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c10">rr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T100" title="ARYTM_3:th.100">ARYTM_3:100</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">eps</font> = <font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font>;<br/><div><span class="lab"><font color="Green" title="E36">A73</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_5_1_2_1"/><span class="lab"><font color="Green" title="E37">A74</font></span>: 
<font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_5_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_5_1_2_1_1"/>
( <font color="Maroon" title="c10">rr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">09</font> or <font color="Maroon" title="c9">3r9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">09</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_5_1_2_1"><span class="lab"><font color="Green" title="E37">A74</font></span></a>, <a class="ref" href="arytm_3.html#T86" title="ARYTM_3:th.86">ARYTM_3:86</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_5_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_5_1_2_1_1_1"/>
<font color="Maroon" title="c10">rr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">09</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:20_5_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">two</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_5_1_2"><span class="lab"><font color="Green" title="E35">A72</font></span></a>, <a class="ref" href="arytm_3.html#T56" title="ARYTM_3:th.56">ARYTM_3:56</a></span>;<br/><span class="kw">hence </span><a NAME="E3:20_5_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_5"><span class="lab"><font color="Green" title="E32">A68</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_5_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_5_1_2_1_1_2"/>
<font color="Maroon" title="c9">3r9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">09</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_5_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_5_1_2"><span class="lab"><font color="Green" title="E34">A71</font></span></a>, <a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5_1_2_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_5_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_5_1_2_2_1"/>
( <font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">r</font> or <font color="Maroon" title="c8">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_5_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_5_1_2_2_1_1"/>
<font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c8">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">t</font> )</span><br/><div class="add"><a NAME="E2:20_5_1_2_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_5_1_2"><span class="lab"><font color="Green" title="E36">A73</font></span></a>, <a class="ref" href="arytm_3.html#T88" title="ARYTM_3:th.88">ARYTM_3:88</a></span>;<br/><a NAME="E3:20_5_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A75</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span></span>)</span> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T83" title="ARYTM_3:th.83">ARYTM_3:83</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c12">t</font> = <font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c8">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c12">t</font> )</span><br/><span class="lab"><font color="Green" title="E38">A76</font></span>: <font color="Maroon" title="c12">t</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">t</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">t</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a></span>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a></span>
<br/>.= 
<font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_5_1_2"><span class="lab"><font color="Green" title="E34">A71</font></span></a>, <a class="ref" href="arytm_3.html#T58" title="ARYTM_3:th.58">ARYTM_3:58</a></span>
<br/>.= 
<font color="Maroon" title="c10">rr</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>
;<br/><a NAME="E6:20_5_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">t</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_5"><span class="lab"><font color="Green" title="E32">A68</font></span></a>, <a class="txt" href="#E5:20_5_1_2"><span class="lab"><font color="Green" title="E35">A72</font></span></a>, <a class="txt" href="#E3:20_5_1_2_2_1_1"><span class="lab"><font color="Green" title="E37">A75</font></span></a>, <a class="txt" href="#E4:20_5_1_2_2_1_1"><span class="lab"><font color="Green" title="E38">A76</font></span></a>, <a class="ref" href="arytm_3.html#T83" title="ARYTM_3:th.83">ARYTM_3:83</a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E7:20_5_1_2_2_1_1"/>
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c12">t</font></span><br/><a NAME="E8:20_5_1_2_2_1_1"/>
<font color="Maroon" title="c5">09</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T71" title="ARYTM_3:th.71">ARYTM_3:71</a></span>;<br/><a NAME="E9:20_5_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">09</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_5_1_2"><span class="lab"><font color="Green" title="E36">A73</font></span></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>;<br/><a NAME="E10:20_5_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c5">09</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T83" title="ARYTM_3:th.83">ARYTM_3:83</a></span>;<br/><span class="kw">hence </span><a NAME="E11:20_5_1_2_2_1_1"/>
<font color="Maroon" title="c8">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c12">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T56" title="ARYTM_3:th.56">ARYTM_3:56</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_5_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_5_1_2_2_1_2"/><span class="lab"><font color="Green" title="E37">A77</font></span>: 
<font color="Maroon" title="c8">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c8">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">t</font> )</span><br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">3r9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">r</font></span>)</span> = 
<font color="Maroon" title="c10">rr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_5_1_2"><span class="lab"><font color="Green" title="E34">A71</font></span></a>, <a class="ref" href="arytm_3.html#T58" title="ARYTM_3:th.58">ARYTM_3:58</a></span>
<br/>.= 
<font color="Maroon" title="c10">rr</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>
;<br/><a NAME="E3:20_5_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A78</font></span>: 
<font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c10">rr</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_5_1_2_2_1_2"><span class="lab"><font color="Green" title="E37">A77</font></span></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c12">t</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">a9</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c7">half</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c8">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c12">t</font> )</span><br/><a NAME="E4:20_5_1_2_2_1_2"/>
<font color="Maroon" title="c1">a9</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>;<br/><a NAME="E5:20_5_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">half</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></span>;<br/><a NAME="E6:20_5_1_2_2_1_2"/><span class="kw">then </span>
<a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c7">half</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="numbers.html#E22"><span class="lab"><font color="Green" title="E14">Lm13</font></span></a>, <a class="ref" href="arytm_3.html#T83" title="ARYTM_3:th.83">ARYTM_3:83</a></span>;<br/><a NAME="E7:20_5_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A79</font></span>: 
<font color="Maroon" title="c12">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_5_1_2"><span class="lab"><font color="Green" title="E33">A70</font></span></a>, <a class="ref" href="arytm_3.html#T88" title="ARYTM_3:th.88">ARYTM_3:88</a></span>;<br/><a NAME="E8:20_5_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A80</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_5_1_2"><span class="lab"><font color="Green" title="E33">A70</font></span></a>, <a class="ref" href="arytm_3.html#T86" title="ARYTM_3:th.86">ARYTM_3:86</a>, <a class="ref" href="arytm_3.html#T88" title="ARYTM_3:th.88">ARYTM_3:88</a></span>;<br/><a NAME="E9:20_5_1_2_2_1_2"/>
<font color="Maroon" title="c1">a9</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c7">half</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T70" title="ARYTM_3:th.70">ARYTM_3:70</a></span>;<br/><a NAME="E10:20_5_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">t</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_5_1_2"><span class="lab"><font color="Green" title="E33">A70</font></span></a>, <a class="txt" href="#E7:20_5_1_2_2_1_2"><span class="lab"><font color="Green" title="E39">A79</font></span></a>, <a class="ref" href="arytm_3.html#T86" title="ARYTM_3:th.86">ARYTM_3:86</a>, <a class="ref" href="arytm_3.html#T88" title="ARYTM_3:th.88">ARYTM_3:88</a></span>;<br/><a NAME="E11:20_5_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A81</font></span>: 
<font color="Maroon" title="c12">t</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20_5_1_2_2_1_2"><span class="lab"><font color="Green" title="E40">A80</font></span></a>, <a class="ref" href="arytm_3.html#T77" title="ARYTM_3:th.77">ARYTM_3:77</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T63" title="ARYTM_3:th.63">ARYTM_3:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T57" title="ARYTM_3:th.57">ARYTM_3:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="numbers.html#E23"><span class="lab"><font color="Green" title="E15">Lm14</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="numbers.html#E23"><span class="lab"><font color="Green" title="E15">Lm14</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T58" title="ARYTM_3:th.58">ARYTM_3:58</a></span>
<br/>.= 
<font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="numbers.html#E23"><span class="lab"><font color="Green" title="E15">Lm14</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T58" title="ARYTM_3:th.58">ARYTM_3:58</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T58" title="ARYTM_3:th.58">ARYTM_3:58</a></span>
;<br/><a NAME="E13:20_5_1_2_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">two</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c2">two</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_5"><span class="lab"><font color="Green" title="E32">A68</font></span></a>, <a class="txt" href="#E5:20_5_1_2"><span class="lab"><font color="Green" title="E35">A72</font></span></a>, <a class="txt" href="#E3:20_5_1_2_2_1_2"><span class="lab"><font color="Green" title="E38">A78</font></span></a>, <a class="ref" href="arytm_3.html#T83" title="ARYTM_3:th.83">ARYTM_3:83</a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></span>;<br/><a NAME="E14:20_5_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">t</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:20_5_1_2_2_1_2"><span class="lab"><font color="Green" title="E41">A81</font></span></a>, <a class="ref" href="arytm_3.html#T76" title="ARYTM_3:th.76">ARYTM_3:76</a></span>;<br/><span class="kw">hence </span><a NAME="E15:20_5_1_2_2_1_2"/>
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c12">t</font></span><br/><a NAME="E16:20_5_1_2_2_1_2"/>
( <font color="Maroon" title="c5">09</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">half</font> &amp; <font color="Maroon" title="c5">09</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c7">half</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a>, <a class="ref" href="arytm_3.html#T71" title="ARYTM_3:th.71">ARYTM_3:71</a></span>;<br/><a NAME="E17:20_5_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">09</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c7">half</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>;<br/><a NAME="E18:20_5_1_2_2_1_2"/><span class="kw">then </span>
<a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c5">09</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c7">half</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T83" title="ARYTM_3:th.83">ARYTM_3:83</a></span>;<br/><a NAME="E19:20_5_1_2_2_1_2"/><span class="kw">then </span>
 <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c7">half</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T56" title="ARYTM_3:th.56">ARYTM_3:56</a></span>;<br/><a NAME="E20:20_5_1_2_2_1_2"/><span class="kw">then </span>
<a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c12">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_5_1_2"><span class="lab"><font color="Green" title="E33">A70</font></span></a>, <a class="ref" href="arytm_3.html#T88" title="ARYTM_3:th.88">ARYTM_3:88</a></span>;<br/><span class="kw">hence </span><a NAME="E21:20_5_1_2_2_1_2"/>
<font color="Maroon" title="c8">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c12">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E8:20_5_1_2"/>
 ex <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c8">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">t</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E23:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A82</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>   st <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/>( ( for <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>   st <font color="Olive" title="b3">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b2">r</font> holds <br/><font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> ) &amp;  ex <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b3">s</font> ) ) </span>}</span> 
 
;<br/>
<a NAME="E24:20"/>
<font color="Maroon" title="c1">a9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c7">half</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">half</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>;<br/>
<a NAME="E25:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A83</font></span>: 
<font color="Maroon" title="c7">half</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E7:20"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E33">A84</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_6"/><span class="lab"><font color="Green" title="E34">A85</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_6_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_6_1"/>
( <font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  or <font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_6"><span class="lab"><font color="Green" title="E34">A85</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_6_1_1"/><span class="lab"><font color="Green" title="E35">A86</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_6_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_6_1_1_1_1"/>
( <font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  ) </span>}</span>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">k</font>,<a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : verum </span>}</span>  or <font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_6_1_1"><span class="lab"><font color="Green" title="E35">A86</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_6_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_6_1_1_1_1_1"/>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  ) </span>}</span>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">k</font>,<a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : verum </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:20_6_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  ) </span>}</span> 
 ;<br/><a NAME="E3:20_6_1_1_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  st <br/>( <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  )
 ;<br/><span class="kw">hence </span><a NAME="E4:20_6_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_6_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_6_1_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A87</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:20_6_1_1_1_1_2"/>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="numbers.html#E16"><span class="lab"><font color="Green" title="E12">Lm11</font></span></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></span>;<br/><a NAME="E3:20_6_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A88</font></span>: 
not <font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_6_1_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_6_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">two</font> or <font color="Maroon" title="c2">two</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_6_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A87</font></span></a>, <a class="txt" href="#E3:20_6_1_1_1_1_2"><span class="lab"><font color="Green" title="E37">A88</font></span></a>, <a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">ORDINAL1:24</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_6_1_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_6_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">two</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:20_6_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">half</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c7">half</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:20"><span class="lab"><font color="Green" title="E32">A83</font></span></a>, <a class="txt" href="numbers.html#E16"><span class="lab"><font color="Green" title="E12">Lm11</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:20_6_1_1_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a>, <a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_6_1_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_6_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c2">two</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:20_6_1_1_1_1_2_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">two</font> &amp; <font color="Olive" title="b1">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font> )
 ;<br/><span class="kw">hence </span><a NAME="E3:20_6_1_1_1_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E29">A65</font></span></a>, <a class="txt" href="numbers.html#E14"><span class="lab"><font color="Green" title="E10">Lm9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:20_6_1_1_1_1_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:20_6_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_6_1_2"/>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:20_6_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:20_6_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_7"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_7"/>
<font color="Maroon" title="c2">two</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:20_7"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Maroon" title="c2">two</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c2">two</font> )
 ;<br/><span class="kw">hence </span><a NAME="E3:20_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E29">A65</font></span></a>, <a class="txt" href="numbers.html#E14"><span class="lab"><font color="Green" title="E10">Lm9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E28:20"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 
;<br/>
<a NAME="E29:20"/><span class="kw">then </span>
not <font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E30:20"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K1" title="ARYTM_2:func.1">DEDEKIND_CUTS</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:20"><span class="lab"><font color="Green" title="E31">A82</font></span></a>, <a class="ref" href="arytm_2.html#D1" title="ARYTM_2:def.1">ARYTM_2:def 1</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E31:20"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="arytm_2.html#K1" title="ARYTM_2:func.1">DEDEKIND_CUTS</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E32:20"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:20"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="arytm_2.html#D2" title="ARYTM_2:def.2">ARYTM_2:def 2</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E33:20"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E34:20"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:20"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E35:20"/>
 <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:20"><span class="lab"><font color="Green" title="E33">A84</font></span></a>, <a class="txt" href="numbers.html#E13"><span class="lab"><font color="Green" title="E9">Lm8</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
