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

<b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>thus </b><a NAME="E1:10_2_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:10_2_2_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:10_2_2_1"/><i><font color="Green">E111</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E3:10_2_2_1"/><i><font color="Green">E112</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E4:10_2_2_1"/><i><font color="Green">E113</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> 1 holds <br/><font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:10_2_2_1_1"/><i><font color="Green">E114</font></i>: 
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E2:10_2_2_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E3:10_2_2_1_1"/><i><font color="Green">E116</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="arytm_0.html#T3" target="_self">Th3</a>, , <a class="ref" href="arytm_2.html#T21">ARYTM_2:21</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:10_2_2_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <a href="arytm_2.html#K2">REAL+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="numbers.html#D1">NUMBERS:def 1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_2_2_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 </span><b>by </b><i><a class="ref" href="arytm_0.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:10_2_2_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/><div class="add"><a NAME="E2:10_2_2_1_1_1_1"/><b>then </b>
 ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> &amp; 1 <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Olive">y'</font> )
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T3" target="_self">Th3</a>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a>, <a class="ref" href="arytm_2.html#T21">ARYTM_2:21</a></i>;<br/><b>hence </b><a NAME="E3:10_2_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="arytm_0.html#T3" target="_self">Th3</a>, <a class="ref" href="arytm_2.html#T4">ARYTM_2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:10_2_2_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><a NAME="E2:10_2_2_1_1_1_2"/><b>then </b>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E3:10_2_2_1_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T3" target="_self">Th3</a>, , <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:10_2_2_1"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E6:10_2_2_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="arytm_0.html#T2" target="_self">Th2</a>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_2_2_1_2"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> ) or ( (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> or  not <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) &amp; (  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> or  not <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> ) ) or ( (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> or  not <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) &amp; (  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> or  not <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_2_1"><b>suppose </b></a><a NAME="E1:10_2_2_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  )
 ;<br/><div class="add"><a NAME="E2:10_2_2_1_2_1"/><b>then </b>
(  ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> &amp; 1 <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Olive">y'</font> ) &amp;  ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> &amp; 1 <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Olive">y'</font> ) )
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T2" target="_self">Th2</a>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>hence </b><a NAME="E3:10_2_2_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_2_2"><b>suppose </b></a><b>that </b><a NAME="E1:10_2_2_1_2_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E2:10_2_2_1_2_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E3:10_2_2_1_2_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E5:10_2_2_1_2_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E6:10_2_2_1_2_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">y'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E7:10_2_2_1_2_2"/><i><font color="Green">E125</font></i>: 
1 <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">x'</font>
 <b>by </b><i>, , <a class="ref" href="arytm_0.html#T4" target="_self">Th4</a>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>consider </b><font color="Maroon">x''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">z'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E9:10_2_2_1_2_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E10:10_2_2_1_2_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">z'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E11:10_2_2_1_2_2"/><i><font color="Green">E128</font></i>: 
1 <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">x''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T2" target="_self">Th2</a>, , <a class="ref" href="arytm_0.html#T5" target="_self">Th5</a>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><a NAME="E12:10_2_2_1_2_2"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E13:10_2_2_1_2_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E14:10_2_2_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, , <a class="ref" href="arytm_0.html#T6" target="_self">Th6</a>, <a class="ref" href="arytm_0.html#T7" target="_self">Th7</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_2_3"><b>suppose </b></a><a NAME="E1:10_2_2_1_2_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 ;<br/><div class="add"><a NAME="E2:10_2_2_1_2_3"/><b>then </b>
 ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Olive">y'</font></span><a href="tarski.html#K4">]</a></span> &amp; 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Olive">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Olive">y'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="arytm_0.html#T1" target="_self">Th1</a>, , <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>hence </b><a NAME="E3:10_2_2_1_2_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_2_4"><b>suppose </b></a><a NAME="E1:10_2_2_1_2_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  )
 ;<br/><div class="add"><a NAME="E2:10_2_2_1_2_4"/><b>then </b>
 ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Olive">x'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> &amp; 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Olive">y'</font> <a href="arytm_2.html#K8">*'</a> <font color="Olive">x'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, , <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>hence </b><a NAME="E3:10_2_2_1_2_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_2_5"><b>suppose </b></a><a NAME="E1:10_2_2_1_2_5"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  )
 ;<br/><div class="add"><a NAME="E2:10_2_2_1_2_5"/><b>then </b>
 ex <font color="Olive">x'</font>, <font color="Olive">z'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Olive">x'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Olive">z'</font> <a href="arytm_2.html#K8">*'</a> <font color="Olive">x'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="arytm_0.html#T2" target="_self">Th2</a>, <a class="ref" href="arytm_0.html#T3" target="_self">Th3</a>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>hence </b><a NAME="E3:10_2_2_1_2_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_2_6"><b>suppose </b></a><a NAME="E1:10_2_2_1_2_6"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 ;<br/><div class="add"><a NAME="E2:10_2_2_1_2_6"/><b>then </b>
 ex <font color="Olive">x'</font>, <font color="Olive">z'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Olive">z'</font></span><a href="tarski.html#K4">]</a></span> &amp; 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Olive">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Olive">z'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="arytm_0.html#T1" target="_self">Th1</a>, <a class="ref" href="arytm_0.html#T2" target="_self">Th2</a>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>hence </b><a NAME="E3:10_2_2_1_2_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_2_7"><b>suppose </b></a><a NAME="E1:10_2_2_1_2_7"/>
( (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> or  not <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) &amp; (  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> or  not <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> ) )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_2_2_1_2_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_2_8"><b>suppose </b></a><a NAME="E1:10_2_2_1_2_8"/>
( (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> or  not <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) &amp; (  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> or  not <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) &amp; (  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> ) )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_2_2_1_2_8"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T2" target="_self">Th2</a>, <a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:10_2_2"/>
( not <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 ;<br/>


</div>
