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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1"/>
( <font color="Maroon" title="c3">A</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="c3">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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c3">A</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"> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <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="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </span>}</span>  is    <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><br/><div class="add"><a NAME="E2:32_1_1_1_1"/>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <font color="Olive" title="b1">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="Olive" title="b2">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b3">s</font></span>)</span> where <font color="Olive" title="b2">r</font>, <font color="Olive" title="b3">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">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E2:32_1_1_1_1_1"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<font color="Maroon" title="c5">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="Olive" title="b1">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <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="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:32_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">r</font>, <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="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1_1_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:32_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_1"><span class="lab"><font color="Green" title="E30">A1</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">hence </span><a NAME="E3:32_1_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <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="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </span>}</span>  is    <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="arytm_2.html#E17"><span class="lab"><font color="Green" title="E11">Lm8</font></span></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c3">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"> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <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="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </span>}</span>  is    <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><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">C</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <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="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </span>}</span> ;<br/><a NAME="E2:32_1_1_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <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="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</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="32_1_1_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"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <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="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </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="E1:32_1_1_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"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <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="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</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="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="E2:32_1_1_1_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">r</font>, <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="b1">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:32_1_1_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><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">C</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <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="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</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="E4:32_1_1_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="c6">C</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="c6">C</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="c6">C</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="32_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</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="c7">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="c6">C</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="c6">C</font> &amp; <font color="Maroon" title="c7">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="E1:32_1_1_1_2_2"/>
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</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="c7">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="c6">C</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="c6">C</font> &amp; <font color="Maroon" title="c7">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">then </span><span class="kw">consider </span><font color="Maroon" title="c8">r0</font>, <font color="Maroon" title="c9">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="E3:32_1_1_1_2_2"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">s0</font>
 <span class="kw">and </span><br/><a NAME="E4:32_1_1_1_2_2"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
<font color="Maroon" title="c8">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E5:32_1_1_1_2_2"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<font color="Maroon" title="c9">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 ;<br/>
<span class="kw">thus </span><a NAME="E6:32_1_1_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="c7">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="c6">C</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="c6">C</font> &amp; <font color="Maroon" title="c7">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="32_1_1_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="c7">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="c6">C</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_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="c7">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="c6">C</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">t0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <span class="kw"> such that </span><br/><a NAME="E3:32_1_1_1_2_2_1"/><span class="lab"><font color="Green" title="E34">A7</font></span>: 
<font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c11">t0</font>
 <span class="kw">and </span><br/><a NAME="E4:32_1_1_1_2_2_1"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
<font color="Maroon" title="c11">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">s0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_1_2_2"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="arytm_3.html#T87" title="ARYTM_3:th.87">ARYTM_3:87</a></span>;<br/>
<a NAME="E5:32_1_1_1_2_2_1"/>
<font color="Maroon" title="c11">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1_1_2_2"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E4:32_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="arytm_2.html#E26"><span class="lab"><font color="Green" title="E20">Lm16</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:32_1_1_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="c6">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_1_2_2"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E3:32_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E34">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"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">consider </span><font color="Maroon" title="c10">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="E8:32_1_1_1_2_2"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
<font color="Maroon" title="c10">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">and </span><br/><a NAME="E9:32_1_1_1_2_2"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
<font color="Maroon" title="c9">s0</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c10">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1_1_2_2"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="arytm_2.html#E16"><span class="lab"><font color="Green" title="E10">Lm7</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_2_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="E1:32_1_1_1_2_2_2"/>
( <font color="Maroon" title="c8">r0</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="c8">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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_2_2_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_2_2_1"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
<font color="Maroon" title="c8">r0</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">  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="c6">C</font> &amp; <font color="Maroon" title="c7">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">s</font> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c11">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="E3:32_1_1_1_2_2_2_1"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<font color="Maroon" title="c11">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E4:32_1_1_1_2_2_2_1"/><span class="lab"><font color="Green" title="E38">A13</font></span>: 
<font color="Maroon" title="c11">r1</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="#E1:32_1_1_1_2"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="arytm_2.html#E25"><span class="lab"><font color="Green" title="E19">Lm15</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">t0</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">r1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Maroon" title="c7">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">t0</font> )</span><br/><span class="kw">thus </span><a NAME="E5:32_1_1_1_2_2_2_1"/>
<font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_1_1_1_2_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E3:32_1_1_1_2_2_2_1"><span class="lab"><font color="Green" title="E37">A12</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="c7">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">t0</font></span><br/><a NAME="E6:32_1_1_1_2_2_2_1"/>
<font color="Maroon" title="c10">t0</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="#E9:32_1_1_1_2_2"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="arytm_3.html#T71" title="ARYTM_3:th.71">ARYTM_3:71</a></span>;<br/><a NAME="E7:32_1_1_1_2_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A14</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="c10">t0</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="#E4:32_1_1_1_2_2_2_1"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="ref" href="arytm_3.html#T86" title="ARYTM_3:th.86">ARYTM_3:86</a></span>;<br/><a NAME="E8:32_1_1_1_2_2_2_1"/><span class="lab"><font color="Green" title="E40">A15</font></span>: 
<font color="Maroon" title="c7">r</font> <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="txt" href="#E3:32_1_1_1_2_2"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E1:32_1_1_1_2_2_2_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></span>;<br/><a NAME="E9:32_1_1_1_2_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T71" title="ARYTM_3:th.71">ARYTM_3:71</a></span>;<br/><span class="kw">hence </span><a NAME="E10:32_1_1_1_2_2_2_1"/>
<font color="Maroon" title="c7">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_1_1_1_2_2_2_1"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="txt" href="#E7:32_1_1_1_2_2_2_1"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_2_2_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_2_2_2"/><span class="lab"><font color="Green" title="E36">A16</font></span>: 
<font color="Maroon" title="c8">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> 
 ; <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="c6">C</font> &amp; <font color="Maroon" title="c7">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">s</font> )</span><br/><div class="add"><a NAME="E2:32_1_1_1_2_2_2_2"/>
<font color="Maroon" title="c9">s0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_1_1_2_2"><span class="lab"><font color="Green" title="E35">A10</font></span></a></span>;<br/><a NAME="E3:32_1_1_1_2_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A17</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_1_2_2"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E1:32_1_1_1_2_2_2_2"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="ref" href="arytm_3.html#T62" title="ARYTM_3:th.62">ARYTM_3:62</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c8">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">t0</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Maroon" title="c7">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">t0</font> )</span><br/><span class="kw">thus </span><a NAME="E4:32_1_1_1_2_2_2_2"/>
<font color="Maroon" title="c8">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">t0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_1_2_2"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E8:32_1_1_1_2_2"><span class="lab"><font color="Green" title="E34">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"> <font color="Maroon" title="c7">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">t0</font></span><br/><a NAME="E5:32_1_1_1_2_2_2_2"/>
<font color="Maroon" title="c7">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c8">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_1_2_2"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E9:32_1_1_1_2_2"><span class="lab"><font color="Green" title="E35">A10</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">hence </span><a NAME="E6:32_1_1_1_2_2_2_2"/>
<font color="Maroon" title="c7">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c8">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_1_2_2_2_2"><span class="lab"><font color="Green" title="E37">A17</font></span></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E5:32_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A18</font></span>: 
<font color="Maroon" title="c6">C</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="c7">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="E7:32_1_1_1_2"/><span class="lab"><font color="Green" title="E32">A19</font></span>: 
not <font color="Maroon" title="c7">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_2.html#E35"><span class="lab"><font color="Green" title="E29">Lm24</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">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="E9:32_1_1_1_2"/><span class="lab"><font color="Green" title="E33">A20</font></span>: 
not <font color="Maroon" title="c8">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_2.html#E35"><span class="lab"><font color="Green" title="E29">Lm24</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:32_1_1_1_2_3"/>
<font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</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="c9">r1</font>, <font color="Maroon" title="c10">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="E3:32_1_1_1_2_3"/><span class="lab"><font color="Green" title="E34">A21</font></span>: 
<font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c8">s0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c10">s1</font>
 <span class="kw">and </span><br/><a NAME="E4:32_1_1_1_2_3"/><span class="lab"><font color="Green" title="E35">A22</font></span>: 
( <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c10">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 ;<br/><a NAME="E5:32_1_1_1_2_3"/>
( <font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">r1</font> or <font color="Maroon" title="c8">s0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c10">s1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_1_2_3"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="ref" href="arytm_3.html#T91" title="ARYTM_3:th.91">ARYTM_3:91</a></span>;<br/><span class="kw">hence </span><a NAME="E6:32_1_1_1_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_1_2"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="txt" href="#E9:32_1_1_1_2"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="txt" href="#E4:32_1_1_1_2_3"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="txt" href="arytm_2.html#E26"><span class="lab"><font color="Green" title="E20">Lm16</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="E11:32_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">C</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">hence </span><a NAME="E12:32_1_1_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <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="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) </span>}</span>  is    <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="#E5:32_1_1_1_2"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</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>
