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

<span class="kw">let </span><font color="Maroon" title="c4">A</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arytm_2.html#K1" title="ARYTM_2:func.1">DEDEKIND_CUTS</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="c4">A</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>  implies  ex <font color="Olive" title="b1">B</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arytm_2.html#K1" title="ARYTM_2:func.1">DEDEKIND_CUTS</a>  st <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">rone</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:61"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
<font color="Maroon" title="c4">A</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> 
 ; <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">B</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arytm_2.html#K1" title="ARYTM_2:func.1">DEDEKIND_CUTS</a>  st <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">rone</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1"/>
( <font color="Maroon" title="c4">A</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="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  </span>}</span>  or  not <font color="Maroon" title="c4">A</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="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  </span>}</span>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_1_1"/>
<font color="Maroon" title="c4">A</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="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  </span>}</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">B</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arytm_2.html#K1" title="ARYTM_2:func.1">DEDEKIND_CUTS</a>  st <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">rone</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">r0</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="E3:61_1_1"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
<font color="Maroon" title="c4">A</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="c5">r0</font> </span>}</span> 
 <span class="kw">and </span><br/><a NAME="E4:61_1_1"/><span class="lab"><font color="Green" title="E58">A3</font></span>: 
<font color="Maroon" title="c5">r0</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">consider </span><font color="Maroon" title="c6">s0</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="E6:61_1_1"/><span class="lab"><font color="Green" title="E59">A4</font></span>: 
<font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s0</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="#E4:61_1_1"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="arytm_3.html#T60" title="ARYTM_3:th.60">ARYTM_3:60</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c7">B</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="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">s0</font> </span>}</span> ;<br/><a NAME="E7:61_1_1"/>
<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="c6">s0</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <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="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c6">s0</font> </span>}</span>  or <font color="Maroon" title="c8">e</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>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_1_1_1"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c6">s0</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="c8">e</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> </span><br/>

<a NAME="E2:61_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">e</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="c6">s0</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:61_1_1_1"/>
<font color="Maroon" title="c8">e</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"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">B</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="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">s0</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>  ;<br/><a NAME="E9:61_1_1"/>
for <font color="Olive" title="b1">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="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> holds <br/>( ( for <font color="Olive" title="b2">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="b2">s</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">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> ) &amp;  ex <font color="Olive" title="b2">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="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Olive" title="b1">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b2">s</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> implies ( ( for <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 <font color="Olive" title="b1">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">r</font> holds <br/><font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> ) &amp;  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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c9">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">s</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_1_1_2"/>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</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">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="b1">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">r</font> holds <br/><font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> ) &amp;  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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c9">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">s</font> ) )</span><br/>

<a NAME="E2:61_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A5</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="c9">r</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="c6">s0</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">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> <span class="kw"> such that </span><br/><a NAME="E4:61_1_1_2"/><span class="lab"><font color="Green" title="E61">A6</font></span>: 
<font color="Maroon" title="c9">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c10">t</font>
 <span class="kw">and </span><br/><a NAME="E5:61_1_1_2"/><span class="lab"><font color="Green" title="E62">A7</font></span>: 
<font color="Maroon" title="c10">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">s0</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/>
<span class="kw">thus </span><a NAME="E6:61_1_1_2"/>
for <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 <font color="Olive" title="b1">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">r</font> holds <br/><font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</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">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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c9">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">s</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">s</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="c11">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">r</font> implies <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_1_1_2_1"/>
<font color="Maroon" title="c11">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">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="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font></span><br/>

<a NAME="E2:61_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">s0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_1_2"><span class="lab"><font color="Green" title="E60">A5</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="E3:61_1_1_2_1"/>
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</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>

<span class="kw">take </span>
<font color="Maroon" title="c10">t</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="c10">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c9">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c10">t</font> )</span><br/>

<span class="kw">thus </span><a NAME="E7:61_1_1_2"/>
<font color="Maroon" title="c10">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_1_2"><span class="lab"><font color="Green" title="E62">A7</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="c9">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c10">t</font></span><br/>

<span class="kw">thus </span><a NAME="E8:61_1_1_2"/>
<font color="Maroon" title="c9">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c10">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_1_2"><span class="lab"><font color="Green" title="E61">A6</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="E10:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A8</font></span>: 
<font color="Maroon" title="c8">B</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="E11:61_1_1"/>
for <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>   holds <br/>(  not <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s0</font> or  not <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">s0</font> )
 ;<br/><a NAME="E12:61_1_1"/><span class="kw">then </span>
not <font color="Maroon" title="c6">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 ;<br/><a NAME="E13:61_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">B</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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">B</font> = <font color="Maroon" title="c8">B</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <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="#E10:61_1_1"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/><a NAME="E15:61_1_1"/><span class="lab"><font color="Green" title="E61">A9</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c9">B</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="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s0</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:61_1_1_3"/>
<font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c9">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s0</font> </span>}</span> 
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s0</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c9">B</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <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="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c9">B</font> or <font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s0</font> </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_1_1_3_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c9">B</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="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s0</font> </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">r1</font>, <font color="Maroon" title="c12">s1</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="E3:61_1_1_3_1"/><span class="lab"><font color="Green" title="E62">A10</font></span>: 
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s1</font>
 <span class="kw">and </span><br/><a NAME="E4:61_1_1_3_1"/><span class="lab"><font color="Green" title="E63">A11</font></span>: 
<font color="Maroon" title="c11">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E5:61_1_1_3_1"/><span class="lab"><font color="Green" title="E64">A12</font></span>: 
<font color="Maroon" title="c12">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 ;<br/>
<a NAME="E6:61_1_1_3_1"/>
 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="c11">r1</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="c5">r0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="txt" href="#E4:61_1_1_3_1"><span class="lab"><font color="Green" title="E63">A11</font></span></a></span>;<br/>
<a NAME="E7:61_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A13</font></span>: 
<font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s1</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s1</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="E8:61_1_1_3_1"/><span class="lab"><font color="Green" title="E66">A14</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="c12">s1</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="c6">s0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_1_3_1"><span class="lab"><font color="Green" title="E64">A12</font></span></a></span>;<br/>
<a NAME="E9:61_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">s0</font>
 
;<br/>
<a NAME="E10:61_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A15</font></span>: 
<font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_1"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="arytm_3.html#T62" title="ARYTM_3:th.62">ARYTM_3:62</a></span>;<br/>
<a NAME="E11:61_1_1_3_1"/>
<font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s1</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61_1_1_3_1"><span class="lab"><font color="Green" title="E66">A14</font></span></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></span>;<br/>
<a NAME="E12:61_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s1</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:61_1_1_3_1"><span class="lab"><font color="Green" title="E67">A15</font></span></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>;<br/>
<a NAME="E13:61_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s1</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61_1_1_3_1"><span class="lab"><font color="Green" title="E65">A13</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="E14:61_1_1_3_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s0</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1_3_1"><span class="lab"><font color="Green" title="E62">A10</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>

<span class="kw">let </span><font color="Maroon" title="c10">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <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="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s0</font> </span>}</span>  or <font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c9">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:61_1_1_3"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s0</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="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c9">B</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">s1</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="E4:61_1_1_3"/><span class="lab"><font color="Green" title="E62">A16</font></span>: 
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s1</font>
 <span class="kw">and </span><br/><a NAME="E5:61_1_1_3"/><span class="lab"><font color="Green" title="E63">A17</font></span>: 
<font color="Maroon" title="c11">s1</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s0</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">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="E7:61_1_1_3"/><span class="lab"><font color="Green" title="E64">A18</font></span>: 
<font color="Maroon" title="c11">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">t0</font>
 <span class="kw">and </span><br/><a NAME="E8:61_1_1_3"/><span class="lab"><font color="Green" title="E65">A19</font></span>: 
<font color="Maroon" title="c12">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c6">s0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_1_3"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="ref" href="arytm_3.html#T87" title="ARYTM_3:th.87">ARYTM_3:87</a></span>;<br/>
<a NAME="E9:61_1_1_3"/>
<font color="Maroon" title="c12">t0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">s0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_1_3"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="txt" href="#E7:61_1_1_3"><span class="lab"><font color="Green" title="E64">A18</font></span></a></span>;<br/>
<a NAME="E10:61_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">t0</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">s0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61_1_1_3"><span class="lab"><font color="Green" title="E65">A19</font></span></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>;<br/>
<a NAME="E11:61_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">t1</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="E13:61_1_1_3"/><span class="lab"><font color="Green" title="E66">A20</font></span>: 
<font color="Maroon" title="c13">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 <span class="kw">and </span><br/><a NAME="E14:61_1_1_3"/><span class="lab"><font color="Green" title="E67">A21</font></span>: 
<font color="Maroon" title="c12">t0</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c13">t1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_2.html#E16"><span class="lab"><font color="Green" title="E10">Lm7</font></span></a></span>;<br/>
<a NAME="E15:61_1_1_3"/>
<font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c13">t1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">r0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:61_1_1_3"><span class="lab"><font color="Green" title="E67">A21</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">then </span><span class="kw">consider </span><font color="Maroon" title="c14">r1</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="E17:61_1_1_3"/><span class="lab"><font color="Green" title="E68">A22</font></span>: 
<font color="Maroon" title="c5">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">r1</font>
 <span class="kw">and </span><br/><a NAME="E18:61_1_1_3"/><span class="lab"><font color="Green" title="E69">A23</font></span>: 
<font color="Maroon" title="c14">r1</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T87" title="ARYTM_3:th.87">ARYTM_3:87</a></span>;<br/>
<a NAME="E19:61_1_1_3"/>
<font color="Maroon" title="c12">t0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">t1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:61_1_1_3"><span class="lab"><font color="Green" title="E67">A21</font></span></a></span>;<br/>
<a NAME="E20:61_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">r0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_1"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E17:61_1_1_3"><span class="lab"><font color="Green" title="E68">A22</font></span></a>, <a class="ref" href="arytm_3.html#T62" title="ARYTM_3:th.62">ARYTM_3:62</a></span>;<br/>
<a NAME="E21:61_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">r1</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">r0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:61_1_1_3"><span class="lab"><font color="Green" title="E69">A23</font></span></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>;<br/>
<a NAME="E22:61_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E57">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:61_1_1_3"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c9">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_1_3"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="txt" href="#E7:61_1_1_3"><span class="lab"><font color="Green" title="E64">A18</font></span></a>, <a class="txt" href="#E13:61_1_1_3"><span class="lab"><font color="Green" title="E66">A20</font></span></a>, <a class="txt" href="#E17:61_1_1_3"><span class="lab"><font color="Green" title="E68">A22</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="E16:61_1_1"/>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Olive" title="b1">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">rone</font> &amp;  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">rone</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<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">t0</font> </span>}</span>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_2.html#D3" target="_self" title="ARYTM_2:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E17:61_1_1"/>
 ex <font color="Olive" title="b1">B</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arytm_2.html#K1" title="ARYTM_2:func.1">DEDEKIND_CUTS</a>  st <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">rone</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61_1_1"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E15:61_1_1"><span class="lab"><font color="Green" title="E61">A9</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="61_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_1_2"/><span class="lab"><font color="Green" title="E57">A24</font></span>: 
not <font color="Maroon" title="c4">A</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="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  </span>}</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">B</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arytm_2.html#K1" title="ARYTM_2:func.1">DEDEKIND_CUTS</a>  st <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">rone</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y9</font> where <font color="Olive" title="b1">y9</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>  :  ex <font color="Olive" title="b2">x9</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/>( not <font color="Olive" title="b2">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">y9</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>}</span> ;<br/><a NAME="E2:61_1_2"/><span class="lab"><font color="Green" title="E58">A25</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">y9</font> where <font color="Olive" title="b1">y9</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>  :  ex <font color="Olive" title="b2">x9</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/>( not <font color="Olive" title="b2">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">y9</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>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <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">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y9</font> where <font color="Olive" title="b1">y9</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>  :  ex <font color="Olive" title="b2">x9</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/>( not <font color="Olive" title="b2">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">y9</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>}</span>  or <font color="Maroon" title="c6">e</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>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_1_2_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y9</font> where <font color="Olive" title="b1">y9</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>  :  ex <font color="Olive" title="b2">x9</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/>( not <font color="Olive" title="b2">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">y9</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>}</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="c6">e</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> </span><br/>

<a NAME="E2:61_1_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y9</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">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e</font> &amp;  ex <font color="Olive" title="b2">x9</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/>( not <font color="Olive" title="b2">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">y9</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>  ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:61_1_2_1"/>
<font color="Maroon" title="c6">e</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"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:61_1_2"/><span class="lab"><font color="Green" title="E59">A26</font></span>: 
<font color="Maroon" title="c4">A</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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x0</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="E5:61_1_2"/><span class="lab"><font color="Green" title="E60">A27</font></span>: 
not <font color="Maroon" title="c6">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T49" title="SUBSET_1:th.49">SUBSET_1:49</a></span>;<br/><a NAME="E6:61_1_2"/>
<font color="Maroon" title="c6">x0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></span>;<br/><a NAME="E7:61_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">x0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <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#T71" title="ARYTM_3:th.71">ARYTM_3:71</a></span>;<br/><a NAME="E8:61_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A28</font></span>: 
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y9</font> where <font color="Olive" title="b1">y9</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>  :  ex <font color="Olive" title="b2">x9</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/>( not <font color="Olive" title="b2">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">y9</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>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_2"><span class="lab"><font color="Green" title="E60">A27</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y9</font> where <font color="Olive" title="b1">y9</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>  :  ex <font color="Olive" title="b2">x9</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/>( not <font color="Olive" title="b2">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">y9</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>}</span>  as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E58">A25</font></span></a></span>;<br/><a NAME="E10:61_1_2"/><span class="lab"><font color="Green" title="E62">A29</font></span>: 
<font color="Maroon" title="c4">A</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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/><a NAME="E11:61_1_2"/>
 ex <font color="Olive" title="b1">x9</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">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></span>;<br/><a NAME="E12:61_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A30</font></span>: 
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:61_1_2"><span class="lab"><font color="Green" title="E62">A29</font></span></a>, <a class="txt" href="arytm_2.html#E48"><span class="lab"><font color="Green" title="E38">Lm29</font></span></a>, <a class="ref" href="arytm_3.html#T71" title="ARYTM_3:th.71">ARYTM_3:71</a></span>;<br/><a NAME="E13:61_1_2"/>
for <font color="Olive" title="b1">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="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> holds <br/>( ( for <font color="Olive" title="b2">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="b2">s</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">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> ) &amp;  ex <font color="Olive" title="b2">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="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Olive" title="b1">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b2">s</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_2"><span class="kw">proof </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="c7">B</font> implies ( ( for <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 <font color="Olive" title="b1">s</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">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> ) &amp;  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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</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">s</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_1_2_2"/>
<font color="Maroon" title="c8">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</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">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="b1">s</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">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> ) &amp;  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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</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">s</font> ) )</span><br/>

<a NAME="E2:61_1_2_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y9</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">y9</font> &amp;  ex <font color="Olive" title="b2">x9</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/>( not <font color="Olive" title="b2">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">y9</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>  ) )
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x9</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="E4:61_1_2_2"/><span class="lab"><font color="Green" title="E64">A31</font></span>: 
not <font color="Maroon" title="c9">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E5:61_1_2_2"/><span class="lab"><font color="Green" title="E65">A32</font></span>: 
<font color="Maroon" title="c9">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</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> 
 ;<br/>
<span class="kw">thus </span><a NAME="E6:61_1_2_2"/>
for <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 <font color="Olive" title="b1">s</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">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</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">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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</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">s</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">s</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="c10">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">r</font> implies <font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_1_2_2_1"/>
<font color="Maroon" title="c10">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&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"> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font></span><br/>

<a NAME="E2:61_1_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">x9</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#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></span>;<br/>
<a NAME="E3:61_1_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">s</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="#E5:61_1_2_2"><span class="lab"><font color="Green" title="E65">A32</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="E4:61_1_2_2_1"/>
<font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_2_2"><span class="lab"><font color="Green" title="E64">A31</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="E7:61_1_2_2"/>
<font color="Maroon" title="c4">A</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>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_2"><span class="lab"><font color="Green" title="E57">A24</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">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x99</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="E9:61_1_2_2"/><span class="lab"><font color="Green" title="E66">A33</font></span>: 
not <font color="Maroon" title="c10">x99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E10:61_1_2_2"/><span class="lab"><font color="Green" title="E67">A34</font></span>: 
<font color="Maroon" title="c10">x99</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c9">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E4:61_1_2_2"><span class="lab"><font color="Green" title="E64">A31</font></span></a>, <a class="txt" href="arytm_2.html#E28"><span class="lab"><font color="Green" title="E22">Lm18</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">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="E12:61_1_2_2"/><span class="lab"><font color="Green" title="E68">A35</font></span>: 
 <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x99</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c11">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61_1_2"><span class="lab"><font color="Green" title="E63">A30</font></span></a>, <a class="txt" href="#E9:61_1_2_2"><span class="lab"><font color="Green" title="E66">A33</font></span></a>, <a class="ref" href="arytm_3.html#T61" title="ARYTM_3:th.61">ARYTM_3:61</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">s</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="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</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="c11">s</font> )</span><br/>

<a NAME="E13:61_1_2_2"/>
<font color="Maroon" title="c10">x99</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c11">s</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="#E12:61_1_2_2"><span class="lab"><font color="Green" title="E68">A35</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:61_1_2_2"/>
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_1_2_2"><span class="lab"><font color="Green" title="E66">A33</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="c11">s</font></span><br/>

<a NAME="E15:61_1_2_2"/><span class="lab"><font color="Green" title="E69">A36</font></span>: 
<font color="Maroon" title="c11">s</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61_1_2_2"><span class="lab"><font color="Green" title="E68">A35</font></span></a>, <a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></span>;<br/>
<a NAME="E16:61_1_2_2"/>
<font color="Maroon" title="c10">x99</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">x9</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="#E10:61_1_2_2"><span class="lab"><font color="Green" title="E67">A34</font></span></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></span>;<br/>
<a NAME="E17:61_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">x99</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <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">x99</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c11">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_2_2"><span class="lab"><font color="Green" title="E65">A32</font></span></a>, <a class="txt" href="#E12:61_1_2_2"><span class="lab"><font color="Green" title="E68">A35</font></span></a>, <a class="ref" href="arytm_3.html#T74" title="ARYTM_3:th.74">ARYTM_3:74</a></span>;<br/>
<a NAME="E18:61_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A37</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="c11">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61_1_2"><span class="lab"><font color="Green" title="E63">A30</font></span></a>, <a class="txt" href="#E9:61_1_2_2"><span class="lab"><font color="Green" title="E66">A33</font></span></a>, <a class="ref" href="arytm_3.html#T88" title="ARYTM_3:th.88">ARYTM_3:88</a></span>;<br/>
<a NAME="E19:61_1_2_2"/>
( <font color="Maroon" title="c10">x99</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">x9</font> &amp; <font color="Maroon" title="c10">x99</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c11">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c11">s</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:61_1_2_2"><span class="lab"><font color="Green" title="E67">A34</font></span></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></span>;<br/>
<a NAME="E20:61_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_2_2"><span class="lab"><font color="Green" title="E65">A32</font></span></a>, <a class="txt" href="#E12:61_1_2_2"><span class="lab"><font color="Green" title="E68">A35</font></span></a>, <a class="txt" href="#E15:61_1_2_2"><span class="lab"><font color="Green" title="E69">A36</font></span></a>, <a class="ref" href="arytm_3.html#T62" title="ARYTM_3:th.62">ARYTM_3:62</a>, <a class="ref" href="arytm_3.html#T73" title="ARYTM_3:th.73">ARYTM_3:73</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:61_1_2_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="c11">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:61_1_2_2"><span class="lab"><font color="Green" title="E70">A37</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><a NAME="E14:61_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A38</font></span>: 
<font color="Maroon" title="c7">B</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/><span class="kw">consider </span><font color="Maroon" title="c8">x9</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="E16:61_1_2"/><span class="lab"><font color="Green" title="E65">A39</font></span>: 
<font color="Maroon" title="c8">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E17:61_1_2"/><span class="lab"><font color="Green" title="E66">A40</font></span>: 
<font color="Maroon" title="c8">x9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E10:61_1_2"><span class="lab"><font color="Green" title="E62">A29</font></span></a>, <a class="txt" href="arytm_2.html#E69"><span class="lab"><font color="Green" title="E54">Lm42</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">y9</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="E19:61_1_2"/><span class="lab"><font color="Green" title="E67">A41</font></span>: 
<font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">y9</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="#E17:61_1_2"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="ref" href="arytm_3.html#T61" title="ARYTM_3:th.61">ARYTM_3:61</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:61_1_2_3"/>
<font color="Maroon" title="c9">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</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="E2:61_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A42</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="c9">y9</font> &amp;  ex <font color="Olive" title="b2">x9</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/>( not <font color="Olive" title="b2">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x9</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#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  ) )
 ;<br/><a NAME="E3:61_1_2_3"/>
<font color="Maroon" title="c9">y9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E19:61_1_2"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></span>;<br/><span class="kw">hence </span><a NAME="E4:61_1_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:61_1_2"><span class="lab"><font color="Green" title="E62">A29</font></span></a>, <a class="txt" href="#E16:61_1_2"><span class="lab"><font color="Green" title="E65">A39</font></span></a>, <a class="txt" href="#E19:61_1_2"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="txt" href="#E2:61_1_2_3"><span class="lab"><font color="Green" title="E68">A42</font></span></a>, <a class="txt" href="arytm_2.html#E48"><span class="lab"><font color="Green" title="E38">Lm29</font></span></a>, <a class="ref" href="arytm_3.html#T88" title="ARYTM_3:th.88">ARYTM_3:88</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="E21:61_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">B</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="E22:61_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c7">B</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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">B</font> = <font color="Maroon" title="c7">B</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <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="#E14:61_1_2"><span class="lab"><font color="Green" title="E64">A38</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">take </span>
<font color="Maroon" title="c10">B</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="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c10">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">rone</font></span><br/><a NAME="E24:61_1_2"/>
for <font color="Olive" title="b1">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>  holds <br/> ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c10">B</font> iff <font color="Olive" title="b1">r</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>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">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="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c10">B</font> iff <font color="Maroon" title="c11">r</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><br/>

<div><span class="kw">hereby </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="c11">r</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>  implies <font color="Maroon" title="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c10">B</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:61_1_2_4_1"/>
<font color="Maroon" title="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c10">B</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="c11">r</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><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">s</font>, <font color="Maroon" title="c13">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> <span class="kw"> such that </span><br/><a NAME="E3:61_1_2_4_1"/><span class="lab"><font color="Green" title="E68">A43</font></span>: 
<font color="Maroon" title="c11">r</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="c13">t</font>
 <span class="kw">and </span><br/><a NAME="E4:61_1_2_4_1"/><span class="lab"><font color="Green" title="E69">A44</font></span>: 
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E5:61_1_2_4_1"/><span class="lab"><font color="Green" title="E70">A45</font></span>: 
<font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 ;<br/><a NAME="E6:61_1_2_4_1"/>
 ex <font color="Olive" title="b1">z9</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">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t</font> &amp;  ex <font color="Olive" title="b2">x9</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/>( not <font color="Olive" title="b2">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">z9</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="#E5:61_1_2_4_1"><span class="lab"><font color="Green" title="E70">A45</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">x9</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="E8:61_1_2_4_1"/><span class="lab"><font color="Green" title="E71">A46</font></span>: 
not <font color="Maroon" title="c14">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E9:61_1_2_4_1"/><span class="lab"><font color="Green" title="E72">A47</font></span>: 
<font color="Maroon" title="c14">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c13">t</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> 
 ;<br/><a NAME="E10:61_1_2_4_1"/>
<font color="Maroon" title="c12">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c14">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:61_1_2"><span class="lab"><font color="Green" title="E62">A29</font></span></a>, <a class="txt" href="#E4:61_1_2_4_1"><span class="lab"><font color="Green" title="E69">A44</font></span></a>, <a class="txt" href="#E8:61_1_2_4_1"><span class="lab"><font color="Green" title="E71">A46</font></span></a>, <a class="txt" href="arytm_2.html#E48"><span class="lab"><font color="Green" title="E38">Lm29</font></span></a></span>;<br/><a NAME="E11:61_1_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A48</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="c13">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c14">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c13">t</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/><div><span class="lab"><font color="Green" title="E74">A49</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_4_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:61_1_2_4_1_1"/><span class="lab"><font color="Green" title="E75">A50</font></span>: 
<font color="Maroon" title="c11">r</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 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="E2:61_1_2_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">t</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2_4_1"><span class="lab"><font color="Green" title="E68">A43</font></span></a>, <a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></span>;<br/><span class="kw">hence </span><a NAME="E3:61_1_2_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2_4_1"><span class="lab"><font color="Green" title="E68">A43</font></span></a>, <a class="txt" href="#E4:61_1_2_4_1"><span class="lab"><font color="Green" title="E69">A44</font></span></a>, <a class="txt" href="#E8:61_1_2_4_1"><span class="lab"><font color="Green" title="E71">A46</font></span></a>, <a class="txt" href="#E9:61_1_2_4_1"><span class="lab"><font color="Green" title="E72">A47</font></span></a>, <a class="txt" href="#E11:61_1_2_4_1"><span class="lab"><font color="Green" title="E73">A48</font></span></a>, <a class="txt" href="#E1:61_1_2_4_1_1"><span class="lab"><font color="Green" title="E75">A50</font></span></a>, <a class="ref" href="arytm_3.html#T62" title="ARYTM_3:th.62">ARYTM_3:62</a>, <a class="ref" href="arytm_3.html#T73" title="ARYTM_3:th.73">ARYTM_3:73</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="E13:61_1_2_4_1"/>
<font color="Maroon" title="c11">r</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="#E3:61_1_2_4_1"><span class="lab"><font color="Green" title="E68">A43</font></span></a>, <a class="txt" href="#E9:61_1_2_4_1"><span class="lab"><font color="Green" title="E72">A47</font></span></a>, <a class="txt" href="#E11:61_1_2_4_1"><span class="lab"><font color="Green" title="E73">A48</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="E14:61_1_2_4_1"/>
<font color="Maroon" title="c11">r</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="#E12:61_1_2_4_1"><span class="lab"><font color="Green" title="E74">A49</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>

<span class="kw">assume </span><a NAME="E2:61_1_2_4"/><span class="lab"><font color="Green" title="E68">A51</font></span>: 
<font color="Maroon" title="c11">r</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"> <font color="Maroon" title="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c10">B</font></span><br/>

<a NAME="E3:61_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A52</font></span>: 
<font color="Maroon" title="c11">r</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> 
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_4_2"><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="E1:61_1_2_4_2"/>
( <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  or <font color="Maroon" title="c11">r</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_4_2_1"><span class="kw">suppose </span></a><a NAME="E1:61_1_2_4_2_1"/>
<font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</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="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c10">B</font></span><br/><div class="add"><a NAME="E2:61_1_2_4_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</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="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></span>;<br/><span class="kw">hence </span><a NAME="E3:61_1_2_4_2_1"/>
<font color="Maroon" title="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c10">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61_1_2"><span class="lab"><font color="Green" title="E61">A28</font></span></a>, <a class="txt" href="#E12:61_1_2"><span class="lab"><font color="Green" title="E63">A30</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="61_1_2_4_2_2"><span class="kw">suppose </span></a><a NAME="E1:61_1_2_4_2_2"/>
<font color="Maroon" title="c11">r</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> 
 ; <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="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c10">B</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">r9</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="E3:61_1_2_4_2_2"/><span class="lab"><font color="Green" title="E70">A53</font></span>: 
<font color="Maroon" title="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">r9</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#T61" title="ARYTM_3:th.61">ARYTM_3:61</a></span>;<br/><a NAME="E4:61_1_2_4_2_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">s</font></span>)</span> 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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_4_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <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="c13">e</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="Maroon" title="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">s</font></span>)</span> 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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  or <font color="Maroon" title="c13">e</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>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_1_2_4_2_2_1"/>
<font color="Maroon" title="c13">e</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="Maroon" title="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">s</font></span>)</span> 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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</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="c13">e</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> </span><br/>

<a NAME="E2:61_1_2_4_2_2_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="Maroon" title="c13">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:61_1_2_4_2_2_1"/>
<font color="Maroon" title="c13">e</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"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">rA</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">s</font></span>)</span> 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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</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>  ;<br/><span class="kw">consider </span><font color="Maroon" title="c14">dr</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="E7:61_1_2_4_2_2"/><span class="lab"><font color="Green" title="E71">A54</font></span>: 
<font color="Maroon" title="c11">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c14">dr</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="#E2:61_1_2_4"><span class="lab"><font color="Green" title="E68">A51</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="kw">consider </span><font color="Maroon" title="c15">xx</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="E9:61_1_2_4_2_2"/><span class="lab"><font color="Green" title="E72">A55</font></span>: 
not <font color="Maroon" title="c15">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2"><span class="lab"><font color="Green" title="E59">A26</font></span></a>, <a class="ref" href="subset_1.html#T49" title="SUBSET_1:th.49">SUBSET_1:49</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c16">rr</font> = <font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font>;<br/><a NAME="E10:61_1_2_4_2_2"/>
<font color="Maroon" title="c14">dr</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2_4"><span class="lab"><font color="Green" title="E69">A52</font></span></a>, <a class="txt" href="#E7:61_1_2_4_2_2"><span class="lab"><font color="Green" title="E71">A54</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">then </span><span class="kw">consider </span><font color="Maroon" title="c17">n0</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="E12:61_1_2_4_2_2"/><span class="lab"><font color="Green" title="E73">A56</font></span>: 
<font color="Maroon" title="c17">n0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <span class="kw">and </span><br/><a NAME="E13:61_1_2_4_2_2"/><span class="lab"><font color="Green" title="E74">A57</font></span>: 
<font color="Maroon" title="c15">xx</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c17">n0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:61_1_2"><span class="lab"><font color="Green" title="E66">A40</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#T103" title="ARYTM_3:th.103">ARYTM_3:103</a></span>;<br/><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> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>;<br/><a NAME="E14:61_1_2_4_2_2"/><span class="lab"><font color="Green" title="E75">A58</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61_1_2"><span class="lab"><font color="Green" title="E63">A30</font></span></a>, <a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></span>;<br/><a NAME="E15:61_1_2_4_2_2"/><span class="lab"><font color="Green" title="E76">A59</font></span>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">n0</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:61_1_2"><span class="lab"><font color="Green" title="E62">A29</font></span></a>, <a class="txt" href="#E9:61_1_2_4_2_2"><span class="lab"><font color="Green" title="E72">A55</font></span></a>, <a class="txt" href="#E13:61_1_2_4_2_2"><span class="lab"><font color="Green" title="E74">A57</font></span></a>, <a class="txt" href="arytm_2.html#E48"><span class="lab"><font color="Green" title="E38">Lm29</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">n1</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="E17:61_1_2_4_2_2"/>
<font color="Maroon" title="c18">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <span class="kw">and </span><br/><a NAME="E18:61_1_2_4_2_2"/><span class="lab"><font color="Green" title="E77">A60</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">n1</font>]
 <span class="kw">and </span><br/><a NAME="E19:61_1_2_4_2_2"/><span class="lab"><font color="Green" title="E78">A61</font></span>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> ]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="arytm_3.html#S1" title="ARYTM_3:sch.1">ARYTM_3:sch 1</a>(<a class="txt" href="arytm_2.html#E52"><span class="lab"><font color="Green" title="E41">Lm32</font></span></a>, <a class="txt" href="arytm_2.html#E53"><span class="lab"><font color="Green" title="E42">Lm33</font></span></a>, <a class="txt" href="#E12:61_1_2_4_2_2"><span class="lab"><font color="Green" title="E73">A56</font></span></a>, <a class="txt" href="#E14:61_1_2_4_2_2"><span class="lab"><font color="Green" title="E75">A58</font></span></a>, <a class="txt" href="#E15:61_1_2_4_2_2"><span class="lab"><font color="Green" title="E76">A59</font></span></a>);<br/></span><span class="kw">set </span><font color="Maroon" title="c19">s0</font> = <font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span>;<br/><div><span class="lab"><font color="Green" title="E79">A62</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_4_2_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:61_1_2_4_2_2_2"/>
<font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">rA</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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">s0</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="E3:61_1_2_4_2_2_2"/><span class="lab"><font color="Green" title="E80">A63</font></span>: 
<font color="Maroon" title="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c20">s0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:61_1_2_4_2_2_2"/><span class="lab"><font color="Green" title="E81">A64</font></span>: 
<font color="Maroon" title="c20">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ;<br/><span class="lab"><font color="Green" title="E82">A65</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> </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">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <span class="p1">(<span class="default"><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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</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="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c20">s0</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="c14">dr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">x9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2_4_2_2_2"><span class="lab"><font color="Green" title="E80">A63</font></span></a>, <a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>
;<br/><a NAME="E6:61_1_2_4_2_2_2"/>
( <font color="Maroon" title="c20">s0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">x9</font> or <font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c20">s0</font> )
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">s1</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="E8:61_1_2_4_2_2_2"/><span class="lab"><font color="Green" title="E83">A66</font></span>: 
( ( <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">s0</font> &amp; <font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c21">s1</font> ) or ( <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x9</font> &amp; <font color="Maroon" title="c20">s0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c21">s1</font> ) )
 ;<br/><a NAME="E9:61_1_2_4_2_2_2"/>
<font color="Maroon" title="c14">dr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c14">dr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c21">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61_1_2_4_2_2_2"><span class="lab"><font color="Green" title="E83">A66</font></span></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></span>;<br/><a NAME="E10:61_1_2_4_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A67</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c21">s1</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="c14">dr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">x9</font></span>)</span> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c21">s1</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="c14">dr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c21">s1</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/><a NAME="E11:61_1_2_4_2_2_2"/>
<font color="Maroon" title="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c20">s0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c21">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61_1_2_4_2_2_2"><span class="lab"><font color="Green" title="E83">A66</font></span></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></span>;<br/><a NAME="E12:61_1_2_4_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A68</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c20">s0</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="c14">dr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">x9</font></span>)</span> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c21">s1</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="c14">dr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">x9</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="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c21">s1</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="c14">dr</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c21">s1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c14">dr</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c21">s1</font>
<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="c21">s1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61_1_2_4_2_2"><span class="lab"><font color="Green" title="E71">A54</font></span></a>, <a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>
;<br/><span class="kw">hence </span><a NAME="E14:61_1_2_4_2_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:61_1_2"><span class="lab"><font color="Green" title="E62">A29</font></span></a>, <a class="txt" href="#E16:61_1_2"><span class="lab"><font color="Green" title="E65">A39</font></span></a>, <a class="txt" href="#E19:61_1_2_4_2_2"><span class="lab"><font color="Green" title="E78">A61</font></span></a>, <a class="txt" href="#E4:61_1_2_4_2_2_2"><span class="lab"><font color="Green" title="E81">A64</font></span></a>, <a class="txt" href="#E5:61_1_2_4_2_2_2"><span class="lab"><font color="Green" title="E82">A65</font></span></a>, <a class="txt" href="#E8:61_1_2_4_2_2_2"><span class="lab"><font color="Green" title="E83">A66</font></span></a>, <a class="txt" href="#E12:61_1_2_4_2_2_2"><span class="lab"><font color="Green" title="E85">A68</font></span></a>, <a class="txt" href="#E10:61_1_2_4_2_2_2"><span class="lab"><font color="Green" title="E84">A67</font></span></a>, <a class="txt" href="arytm_2.html#E48"><span class="lab"><font color="Green" title="E38">Lm29</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="lab"><font color="Green" title="E80">A69</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_4_2_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:61_1_2_4_2_2_3"/><span class="lab"><font color="Green" title="E81">A70</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">r9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</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/><font color="Maroon" title="c11">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="c18">n1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">r9</font></span>)</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> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2_4_2_2"><span class="lab"><font color="Green" title="E70">A53</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="c18">n1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</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="kw">hence </span><a NAME="E3:61_1_2_4_2_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:61_1_2_4_2_2"><span class="lab"><font color="Green" title="E79">A62</font></span></a>, <a class="txt" href="#E1:61_1_2_4_2_2_3"><span class="lab"><font color="Green" title="E81">A70</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="E22:61_1_2_4_2_2"/>
<font color="Maroon" title="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></span>;<br/><a NAME="E23:61_1_2_4_2_2"/><span class="kw">then </span>
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">rA</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61_1_2"><span class="lab"><font color="Green" title="E63">A30</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">s9</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="E25:61_1_2_4_2_2"/><span class="lab"><font color="Green" title="E81">A71</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c20">s9</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="#E20:61_1_2_4_2_2"><span class="lab"><font color="Green" title="E79">A62</font></span></a>, <a class="ref" href="arytm_3.html#T61" title="ARYTM_3:th.61">ARYTM_3:61</a></span>;<br/><span class="lab"><font color="Green" title="E82">A72</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c20">s9</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c20">s9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c11">r</font>
<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="c11">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:61_1_2_4_2_2"><span class="lab"><font color="Green" title="E81">A71</font></span></a>, <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"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">r9</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="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c20">s9</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">r9</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c20">s9</font>
<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"><span class="p2">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></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#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c20">s9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2_4_2_2"><span class="lab"><font color="Green" title="E70">A53</font></span></a>, <a class="ref" href="arytm_3.html#T58" title="ARYTM_3:th.58">ARYTM_3:58</a></span>
<br/>.= 
 <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="#E25:61_1_2_4_2_2"><span class="lab"><font color="Green" title="E81">A71</font></span></a>, <a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></span>
;<br/><a NAME="E28:61_1_2_4_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c14">dr</font></span>)</span></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">r9</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="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c20">s9</font></span>)</span> <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> 
 ;<br/><a NAME="E29:61_1_2_4_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c20">s9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:61_1_2_4_2_2"><span class="lab"><font color="Green" title="E80">A69</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E30:61_1_2_4_2_2"/>
<font color="Maroon" title="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c10">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:61_1_2_4_2_2"><span class="lab"><font color="Green" title="E77">A60</font></span></a>, <a class="txt" href="#E26:61_1_2_4_2_2"><span class="lab"><font color="Green" title="E82">A72</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><a NAME="E25:61_1_2"/><span class="kw">then </span>
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K4" title="ARYTM_2:func.4">GLUED</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c10">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">rone</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_2.html#D4" target="_self" title="ARYTM_2:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E26:61_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <font color="Maroon" title="c10">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">rone</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_2.html#E21"><span class="lab"><font color="Green" title="E15">Lm12</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>
