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

<span class="kw">let </span><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> ; <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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Maroon" title="c5">y</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>  implies  ex <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y9</font> &amp; <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="Olive" title="b1">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b2">y9</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:73"/><span class="lab"><font color="Green" title="E60">A1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 <span class="kw">and </span><br/><a NAME="E2:73"/><span class="lab"><font color="Green" title="E61">A2</font></span>: 
<font color="Maroon" title="c5">y</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">  ex <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y9</font> &amp; <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="Olive" title="b1">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b2">y9</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_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="E1:73_1"/>
( <font color="Maroon" title="c4">x</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="c5">y</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="c5">y</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>  &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#K11" title="ARYTM_3:func.11">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1"><span class="kw">suppose </span></a><a NAME="E1:73_1_1"/><span class="lab"><font color="Green" title="E62">A3</font></span>: 
<font color="Maroon" title="c4">x</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">x9</font>, <font color="Olive" title="b2">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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y9</font> &amp; <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="Olive" title="b1">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b2">y9</font> )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c6">y9</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#K5" title="ARYTM_3:func.5">RAT+</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E61">A2</font></span></a></span>;<br/><span class="kw">take </span>
 <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">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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y9</font> &amp; <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> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">y9</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c6">y9</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y9</font> &amp; <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> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">y9</font> )</span><br/><span class="kw">thus </span><a NAME="E3:73_1_1"/>
<font color="Maroon" title="c4">x</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="#E1:73_1_1"><span class="lab"><font color="Green" title="E62">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y9</font> &amp; <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> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">y9</font> )</span><br/><span class="kw">thus </span><a NAME="E4:73_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y9</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">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> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">y9</font></span><br/><span class="kw">thus </span><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#K11" title="ARYTM_3:func.11">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_1"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="arytm_2.html#T4" target="_self" title="ARYTM_2:th.4">Th4</a></span>
<br/>.= 
<a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">y9</font>
<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>
; <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="73_1_2"><span class="kw">suppose </span></a><a NAME="E1:73_1_2"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
<font color="Maroon" title="c5">y</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">x9</font>, <font color="Olive" title="b2">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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y9</font> &amp; <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="Olive" title="b1">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b2">y9</font> )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c6">x9</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#K5" title="ARYTM_3:func.5">RAT+</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E60">A1</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c6">x9</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">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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x9</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y9</font> &amp; <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">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">y9</font> )</span><br/><span class="kw">take </span>
 <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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x9</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  &amp; <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">x9</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>  )</span><br/><span class="kw">thus </span><a NAME="E3:73_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x9</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="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  &amp; <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">x9</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>  )</span><br/><span class="kw">thus </span><a NAME="E4:73_1_2"/>
<font color="Maroon" title="c5">y</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="#E1:73_1_2"><span class="lab"><font color="Green" title="E62">A4</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="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">x9</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> </span><br/><span class="kw">thus </span><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#K11" title="ARYTM_3:func.11">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_2"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="ref" href="arytm_2.html#T4" target="_self" title="ARYTM_2:th.4">Th4</a></span>
<br/>.= 
<font color="Maroon" title="c6">x9</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> 
<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>
; <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="73_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:73_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#K11" title="ARYTM_3:func.11">{}</a> 
 <span class="kw">and </span><br/><a NAME="E2:73_1_3"/><span class="lab"><font color="Green" title="E62">A5</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#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">x9</font>, <font color="Olive" title="b2">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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y9</font> &amp; <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="Olive" title="b1">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b2">y9</font> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c6">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="E4:73_1_3"/><span class="lab"><font color="Green" title="E63">A6</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y9</font>
 <span class="kw">and </span><br/><a NAME="E5:73_1_3"/><span class="lab"><font color="Green" title="E64">A7</font></span>: 
 <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#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">y9</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="arytm_2.html#D3" target="_self" title="ARYTM_2:def.3">Def3</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c7">A</font> =  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">x</font>;<br/><span class="kw">set </span><font color="Maroon" title="c8">B</font> =  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">y</font>;<br/><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="E7:73_1_3"/><span class="lab"><font color="Green" title="E65">A8</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x9</font>
 <span class="kw">and </span><br/><a NAME="E8:73_1_3"/><span class="lab"><font color="Green" title="E66">A9</font></span>: 
 <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#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="c9">x9</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="ref" href="arytm_2.html#D3" target="_self" title="ARYTM_2:def.3">Def3</a></span>;<br/><a NAME="E9:73_1_3"/><span class="lab"><font color="Green" title="E67">A10</font></span>: 
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/> ( <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="c9">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">y9</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#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">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> iff <font color="Maroon" title="c10">s2</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&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="c6">y9</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:73_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="c9">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">y9</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">s2</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&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="c6">y9</font> implies <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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:73_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>
 ; <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">s2</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&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="c6">y9</font></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="E3:73_1_3_1_1"/><span class="lab"><font color="Green" title="E68">A11</font></span>: 
<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#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c12">s1</font>
 <span class="kw">and </span><br/><a NAME="E4:73_1_3_1_1"/><span class="lab"><font color="Green" title="E69">A12</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:73_1_3_1_1"/><span class="lab"><font color="Green" title="E70">A13</font></span>: 
<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="E6:73_1_3_1_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="c9">x9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:73_1_3"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="txt" href="#E4:73_1_3_1_1"><span class="lab"><font color="Green" title="E69">A12</font></span></a></span>;<br/>
<a NAME="E7:73_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">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="c12">s1</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="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="E8:73_1_3_1_1"/><span class="lab"><font color="Green" title="E72">A15</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">y9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_3"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E5:73_1_3_1_1"><span class="lab"><font color="Green" title="E70">A13</font></span></a></span>;<br/>
<a NAME="E9:73_1_3_1_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">y9</font>
 
;<br/>
<a NAME="E10:73_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A16</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="c12">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_3"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E7:73_1_3"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="ref" href="arytm_3.html#T62" title="ARYTM_3:th.62">ARYTM_3:62</a></span>;<br/>
<a NAME="E11:73_1_3_1_1"/>
<font color="Maroon" title="c9">x9</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="c9">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:73_1_3_1_1"><span class="lab"><font color="Green" title="E72">A15</font></span></a>, <a class="ref" href="arytm_3.html#T90" title="ARYTM_3:th.90">ARYTM_3:90</a></span>;<br/>
<a NAME="E12:73_1_3_1_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="c12">s1</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&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="c6">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:73_1_3_1_1"><span class="lab"><font color="Green" title="E73">A16</font></span></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:73_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="c9">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_3_1_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E7:73_1_3_1_1"><span class="lab"><font color="Green" title="E71">A14</font></span></a>, <a class="ref" href="arytm_3.html#T76" title="ARYTM_3:th.76">ARYTM_3:76</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E2:73_1_3_1"/><span class="lab"><font color="Green" title="E68">A17</font></span>: 
<font color="Maroon" title="c10">s2</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&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="c6">y9</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">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></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="E4:73_1_3_1"/><span class="lab"><font color="Green" title="E69">A18</font></span>: 
<font color="Maroon" title="c10">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">t0</font>
 <span class="kw">and </span><br/><a NAME="E5:73_1_3_1"/><span class="lab"><font color="Green" title="E70">A19</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="c6">y9</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="E6:73_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="c6">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_3_1"><span class="lab"><font color="Green" title="E68">A17</font></span></a>, <a class="txt" href="#E4:73_1_3_1"><span class="lab"><font color="Green" title="E69">A18</font></span></a></span>;<br/>
<a NAME="E7:73_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">t0</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_3_1"><span class="lab"><font color="Green" title="E70">A19</font></span></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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="E9:73_1_3_1"/><span class="lab"><font color="Green" title="E71">A20</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E10:73_1_3_1"/><span class="lab"><font color="Green" title="E72">A21</font></span>: 
<font color="Maroon" title="c12">t1</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">y9</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/>
<a NAME="E11:73_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#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c9">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_3_1"><span class="lab"><font color="Green" title="E69">A18</font></span></a>, <a class="txt" href="#E9:73_1_3_1"><span class="lab"><font color="Green" title="E71">A20</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="c13">t2</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="E13:73_1_3_1"/><span class="lab"><font color="Green" title="E73">A22</font></span>: 
<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#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c13">t2</font>
 <span class="kw">and </span><br/><a NAME="E14:73_1_3_1"/><span class="lab"><font color="Green" title="E74">A23</font></span>: 
<font color="Maroon" title="c13">t2</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">x9</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_3_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:73_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="c9">x9</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="E2:73_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_3"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E7:73_1_3"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E4:73_1_3_1"><span class="lab"><font color="Green" title="E69">A18</font></span></a>, <a class="txt" href="#E13:73_1_3_1"><span class="lab"><font color="Green" title="E73">A22</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">hence </span><a NAME="E3:73_1_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_3_1"><span class="lab"><font color="Green" title="E71">A20</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="E16:73_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">t2</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="#E14:73_1_3_1"><span class="lab"><font color="Green" title="E74">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="E17:73_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A24</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:73_1_3"><span class="lab"><font color="Green" title="E66">A9</font></span></a></span>;<br/>
<a NAME="E18:73_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_3"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E10:73_1_3_1"><span class="lab"><font color="Green" title="E72">A21</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:73_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:73_1_3_1"><span class="lab"><font color="Green" title="E73">A22</font></span></a>, <a class="txt" href="#E17:73_1_3_1"><span class="lab"><font color="Green" title="E75">A24</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">then </span><span class="kw">consider </span><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#K5" title="ARYTM_3:func.5">RAT+</a> <span class="kw"> such that </span><br/><a NAME="E11:73_1_3"/><span class="lab"><font color="Green" title="E68">A25</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E12:73_1_3"/><span class="lab"><font color="Green" title="E69">A26</font></span>: 
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/> ( <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> )
 <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">take </span>
<font color="Maroon" title="c9">x9</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">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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x9</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y9</font> &amp; <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="c9">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Olive" title="b1">y9</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c6">y9</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x9</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y9</font> &amp; <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="c9">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">y9</font> )</span><br/><span class="kw">thus </span><a NAME="E13:73_1_3"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x9</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1_3"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E4:73_1_3"><span class="lab"><font color="Green" title="E63">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"> <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="c9">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">y9</font></span><br/><a NAME="E14:73_1_3"/>
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/> ( <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&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="c6">y9</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="73_1_3_2"><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#NR4" title="ARYTM_3:NR.4">&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="c6">y9</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> )</span><br/>

<a NAME="E1:73_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="c9">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">y9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_3"><span class="lab"><font color="Green" title="E67">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:73_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="c9">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">y9</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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:73_1_3"><span class="lab"><font color="Green" title="E69">A26</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="E15:73_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="c9">x9</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:73_1_3"><span class="lab"><font color="Green" title="E68">A25</font></span></a>, <a class="txt" href="arytm_2.html#E14"><span class="lab"><font color="Green" title="E8">Lm6</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>
