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

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:72"/><i><font color="Green" title="E62">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> 
 <b>and </b><br/><a NAME="E2:72"/><i><font color="Green" title="E63">A2</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:72_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  or ( <font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1"><b>suppose </b></a><a NAME="E1:72_1_1"/><i><font color="Green" title="E64">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c6">y'</font> = <font color="Maroon" title="c5">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E2:72"><i><font color="Green" title="E63">A2</font></i></a></i>;<br/><b>take </b>
 <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
;<br/><b>take </b>
<font color="Maroon" title="c6">y'</font>
;<br/><b>thus </b><a NAME="E3:72_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:72_1_1"><i><font color="Green" title="E64">A3</font></i></a></i>;<br/><b>thus </b><a NAME="E4:72_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y'</font>
 ;<br/><b>thus </b><font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c5">y</font> = 
 <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
<b>by </b><i><a class="txt" href="arytm_2.html#E1:72_1_1"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="arytm_2.html#T4" target="_self" title="ARYTM_2:th.4">Th4</a></i>
<br/>.= 
<a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c6">y'</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_2"><b>suppose </b></a><a NAME="E1:72_1_2"/><i><font color="Green" title="E64">A4</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c6">x'</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E1:72"><i><font color="Green" title="E62">A1</font></i></a></i>;<br/><b>take </b>
<font color="Maroon" title="c6">x'</font>
;<br/><b>take </b>
 <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
;<br/><b>thus </b><a NAME="E3:72_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font>
 ;<br/><b>thus </b><a NAME="E4:72_1_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:72_1_2"><i><font color="Green" title="E64">A4</font></i></a></i>;<br/><b>thus </b><font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c5">y</font> = 
 <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
<b>by </b><i><a class="txt" href="arytm_2.html#E1:72_1_2"><i><font color="Green" title="E64">A4</font></i></a>, <a class="ref" href="arytm_2.html#T4" target="_self" title="ARYTM_2:th.4">Th4</a></i>
<br/>.= 
<font color="Maroon" title="c6">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
<b>by </b><i><a class="ref" href="arytm_3.html#T54" title="ARYTM_3:th.54">ARYTM_3:54</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:72_1_3"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 <b>and </b><br/><a NAME="E2:72_1_3"/><i><font color="Green" title="E64">A5</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c6">x'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E4:72_1_3"/><i><font color="Green" title="E65">A6</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font>
 <b>and </b><br/><a NAME="E5:72_1_3"/><i><font color="Green" title="E66">A7</font></i>: 
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">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">x'</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:72"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="arytm_2.html#D3" target="_self" title="ARYTM_2:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">y'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E7:72_1_3"/><i><font color="Green" title="E67">A8</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font>
 <b>and </b><br/><a NAME="E8:72_1_3"/><i><font color="Green" title="E68">A9</font></i>: 
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">y</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#K7" title="ARYTM_3:func.7">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="c7">y'</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:72"><i><font color="Green" title="E63">A2</font></i></a>, <a class="ref" href="arytm_2.html#D3" target="_self" title="ARYTM_2:def.3">Def3</a></i>;<br/><b>take </b>
<font color="Maroon" title="c6">x'</font>
;<br/><b>take </b>
<font color="Maroon" title="c7">y'</font>
;<br/><b>thus </b><a NAME="E9:72_1_3"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> )
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:72_1_3"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="arytm_2.html#E7:72_1_3"><i><font color="Green" title="E67">A8</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c8">A</font> =  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">x</font>;<br/><b>set </b><font color="Maroon" title="c9">B</font> =  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">y</font>;<br/><a NAME="E10:72_1_3"/><i><font color="Green" title="E69">A10</font></i>: 
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#K7" title="ARYTM_3:func.7">RAT+</a>  holds <br/> ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">y</font></span>)</span> iff <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">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c7">y'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">s2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> ;<br/>

<b>thus </b><a NAME="E1:72_1_3_1"/>
( <font color="Maroon" title="c10">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">y</font></span>)</span> implies <font color="Maroon" title="c10">s2</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c7">y'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_1_3_1_1"/>
<font color="Maroon" title="c10">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">y</font></span>)</span>
 ;<br/>

<a NAME="E2:72_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c10">s2</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#K12" title="ARYTM_3:func.12">*'</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#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">y</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><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#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E4:72_1_3_1_1"/><i><font color="Green" title="E70">A11</font></i>: 
<font color="Maroon" title="c10">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">s1</font>
 <b>and </b><br/><a NAME="E5:72_1_3_1_1"/><i><font color="Green" title="E71">A12</font></i>: 
<font color="Maroon" title="c11">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">x</font>
 <b>and </b><br/><a NAME="E6:72_1_3_1_1"/><i><font color="Green" title="E72">A13</font></i>: 
<font color="Maroon" title="c12">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">y</font>
 ;<br/>
<a NAME="E7:72_1_3_1_1"/><i><font color="Green" title="E73">A14</font></i>: 
 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#K7" title="ARYTM_3:func.7">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="c6">x'</font> )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E5:72_1_3"><i><font color="Green" title="E66">A7</font></i></a>, <a class="txt" href="arytm_2.html#E5:72_1_3_1_1"><i><font color="Green" title="E71">A12</font></i></a></i>;<br/>
<a NAME="E8:72_1_3_1_1"/><i><font color="Green" title="E74">A15</font></i>: 
 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#K7" title="ARYTM_3:func.7">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="c7">y'</font> )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E8:72_1_3"><i><font color="Green" title="E68">A9</font></i></a>, <a class="txt" href="arytm_2.html#E6:72_1_3_1_1"><i><font color="Green" title="E72">A13</font></i></a></i>;<br/>
<a NAME="E9:72_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c12">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">y'</font>
 
;<br/>
<a NAME="E10:72_1_3_1_1"/><b>then </b><i><font color="Green" title="E75">A16</font></i>: 
<font color="Maroon" title="c6">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c7">y'</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E2:72_1_3"><i><font color="Green" title="E64">A5</font></i></a>, <a class="txt" href="arytm_2.html#E4:72_1_3"><i><font color="Green" title="E65">A6</font></i></a>, <a class="ref" href="arytm_3.html#T62" title="ARYTM_3:th.62">ARYTM_3:62</a></i>;<br/>
<a NAME="E11:72_1_3_1_1"/>
<font color="Maroon" title="c6">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</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="c6">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c7">y'</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E8:72_1_3_1_1"><i><font color="Green" title="E74">A15</font></i></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></i>;<br/>
<a NAME="E12:72_1_3_1_1"/><b>then </b><i><font color="Green" title="E76">A17</font></i>: 
<font color="Maroon" title="c6">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</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="c6">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c7">y'</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E10:72_1_3_1_1"><i><font color="Green" title="E75">A16</font></i></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></i>;<br/>
<a NAME="E13:72_1_3_1_1"/>
<font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</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="c6">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c12">s1</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E7:72_1_3_1_1"><i><font color="Green" title="E73">A14</font></i></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></i>;<br/>
<b>hence </b><a NAME="E14:72_1_3_1_1"/>
<font color="Maroon" title="c10">s2</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c7">y'</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:72_1_3_1_1"><i><font color="Green" title="E70">A11</font></i></a>, <a class="txt" href="arytm_2.html#E12:72_1_3_1_1"><i><font color="Green" title="E76">A17</font></i></a>, <a class="ref" href="arytm_3.html#T76" title="ARYTM_3:th.76">ARYTM_3:76</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:72_1_3_1"/><i><font color="Green" title="E70">A18</font></i>: 
<font color="Maroon" title="c10">s2</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c7">y'</font>
 ;<br/>

<b>then consider </b><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#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E4:72_1_3_1"/><i><font color="Green" title="E71">A19</font></i>: 
<font color="Maroon" title="c10">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c11">t0</font>
 <b>and </b><br/><a NAME="E5:72_1_3_1"/><i><font color="Green" title="E72">A20</font></i>: 
<font color="Maroon" title="c11">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c7">y'</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T87" title="ARYTM_3:th.87">ARYTM_3:87</a></i>;<br/>
<a NAME="E6:72_1_3_1"/>
<font color="Maroon" title="c11">t0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">y'</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E2:72_1_3_1"><i><font color="Green" title="E70">A18</font></i></a>, <a class="txt" href="arytm_2.html#E4:72_1_3_1"><i><font color="Green" title="E71">A19</font></i></a></i>;<br/>
<a NAME="E7:72_1_3_1"/><b>then </b>
<font color="Maroon" title="c11">t0</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c7">y'</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E5:72_1_3_1"><i><font color="Green" title="E72">A20</font></i></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">t1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E9:72_1_3_1"/><i><font color="Green" title="E73">A21</font></i>: 
<font color="Maroon" title="c11">t0</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c12">t1</font>
 <b>and </b><br/><a NAME="E10:72_1_3_1"/><i><font color="Green" title="E74">A22</font></i>: 
<font color="Maroon" title="c12">t1</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c7">y'</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T101" title="ARYTM_3:th.101">ARYTM_3:101</a></i>;<br/>
<a NAME="E11:72_1_3_1"/>
<font color="Maroon" title="c10">s2</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c12">t1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c6">x'</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E4:72_1_3_1"><i><font color="Green" title="E71">A19</font></i></a>, <a class="txt" href="arytm_2.html#E9:72_1_3_1"><i><font color="Green" title="E73">A21</font></i></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">t2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E13:72_1_3_1"/><i><font color="Green" title="E75">A23</font></i>: 
<font color="Maroon" title="c10">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t1</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c13">t2</font>
 <b>and </b><br/><a NAME="E14:72_1_3_1"/><i><font color="Green" title="E76">A24</font></i>: 
<font color="Maroon" title="c13">t2</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c6">x'</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T87" title="ARYTM_3:th.87">ARYTM_3:87</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_1_3_1_2"/>
<font color="Maroon" title="c13">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font>
 ;<br/><a NAME="E2:72_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c11">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t1</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:72_1_3"><i><font color="Green" title="E64">A5</font></i></a>, <a class="txt" href="arytm_2.html#E4:72_1_3"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="arytm_2.html#E4:72_1_3_1"><i><font color="Green" title="E71">A19</font></i></a>, <a class="txt" href="arytm_2.html#E13:72_1_3_1"><i><font color="Green" title="E75">A23</font></i></a>, <a class="ref" href="arytm_3.html#T62" title="ARYTM_3:th.62">ARYTM_3:62</a></i>;<br/><b>hence </b><a NAME="E3:72_1_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E9:72_1_3_1"><i><font color="Green" title="E73">A21</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:72_1_3_1"/><b>then </b>
<font color="Maroon" title="c13">t2</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">x'</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E14:72_1_3_1"><i><font color="Green" title="E76">A24</font></i></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></i>;<br/>
<a NAME="E17:72_1_3_1"/><b>then </b><i><font color="Green" title="E77">A25</font></i>: 
<font color="Maroon" title="c13">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E5:72_1_3"><i><font color="Green" title="E66">A7</font></i></a></i>;<br/>
<a NAME="E18:72_1_3_1"/>
<font color="Maroon" title="c12">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">y</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E8:72_1_3"><i><font color="Green" title="E68">A9</font></i></a>, <a class="txt" href="arytm_2.html#E10:72_1_3_1"><i><font color="Green" title="E74">A22</font></i></a></i>;<br/>
<a NAME="E19:72_1_3_1"/><b>then </b>
<font color="Maroon" title="c10">s2</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#K12" title="ARYTM_3:func.12">*'</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#K7" title="ARYTM_3:func.7">RAT+</a>  : ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">y</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E13:72_1_3_1"><i><font color="Green" title="E75">A23</font></i></a>, <a class="txt" href="arytm_2.html#E17:72_1_3_1"><i><font color="Green" title="E77">A25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E20:72_1_3_1"/>
<font color="Maroon" title="c10">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">y</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c10">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E12:72_1_3"/><i><font color="Green" title="E70">A26</font></i>: 
 <a href="arytm_2.html#K4" title="ARYTM_2:func.4">GLUED</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <span class="p2">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r</font>
 <b>and </b><br/><a NAME="E13:72_1_3"/><i><font color="Green" title="E71">A27</font></i>: 
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#K7" title="ARYTM_3:func.7">RAT+</a>  holds <br/> ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">y</font></span>)</span> iff <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c10">r</font> )
 <b>by </b><i><a class="ref" href="arytm_2.html#D4" target="_self" title="ARYTM_2:def.4">Def4</a></i>;<br/><a NAME="E14:72_1_3"/><i><font color="Green" title="E72">A28</font></i>: 
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#K7" title="ARYTM_3:func.7">RAT+</a>  holds <br/> ( <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">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c7">y'</font> iff <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c10">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><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#K7" title="ARYTM_3:func.7">RAT+</a> ;<br/>

<a NAME="E1:72_1_3_2"/>
( <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="arytm_2.html#K6" title="ARYTM_2:func.6">*'</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">y</font></span>)</span> iff <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">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c7">y'</font> )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E10:72_1_3"><i><font color="Green" title="E69">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:72_1_3_2"/>
( <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">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c7">y'</font> iff <font color="Maroon" title="c11">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c10">r</font> )
 <b>by </b><i><a class="txt" href="arytm_2.html#E13:72_1_3"><i><font color="Green" title="E71">A27</font></i></a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E15:72_1_3"/>
<font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c7">y'</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E12:72_1_3"><i><font color="Green" title="E70">A26</font></i></a>, <a class="txt" href="arytm_2.html#E14:72_1_3"><i><font color="Green" title="E72">A28</font></i></a>, <a class="txt" href="arytm_2.html#E15"><i><font color="Green" title="E10">Lm7</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
