<?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>assume </b><b>that </b><br/><a NAME="E1:10_1_2"/><i><font color="Green">E44</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E2:10_1_2"/><i><font color="Green">E45</font></i>: 
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_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>  ) 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>  <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 ( <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>  <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>  <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> &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">z</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">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">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> ) &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">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> ) &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> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_1"><b>suppose </b></a><a NAME="E1:10_1_2_1_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_1_2_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; 0 <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> <a href="arytm_2.html#K7">+</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; 0 <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> <a href="arytm_2.html#K7">+</a> <font color="Olive">y'</font> ) )
 <b>by </b><i><a class="ref" href="arytm_0.html#T1" target="_self">Th1</a>, , </i>;<br/><b>hence </b><a NAME="E3:10_1_2_1_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_2.html#T12">ARYTM_2:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:10_1_2_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E2:10_1_2_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E3:10_1_2_1_2"/><i><font color="Green">E52</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_1_2_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E6:10_1_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 <b>and </b><br/><a NAME="E7:10_1_2_1_2"/><i><font color="Green">E54</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y'</font>
 <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>, , </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">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="E9:10_1_2_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>and </b><br/><a NAME="E10:10_1_2_1_2"/><i><font color="Green">E57</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">y''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E11:10_1_2_1_2"/><i><font color="Green">E58</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">y''</font>
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T2" target="_self">Th2</a>, , </i>;<br/><a NAME="E12:10_1_2_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x''</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#T3" target="_self">Th3</a>, , <a class="ref" href="arytm_0.html#T4" target="_self">Th4</a>, <a class="ref" href="arytm_2.html#T6">ARYTM_2:6</a></i>;<br/><a NAME="E13:10_1_2_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E14:10_1_2_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <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/><a NAME="E15:10_1_2_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/><b>hence </b><a NAME="E16:10_1_2_1_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#T5" target="_self">Th5</a>, , , , <a class="ref" href="arytm_1.html#T19">ARYTM_1:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:10_1_2_1_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E2:10_1_2_1_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E3:10_1_2_1_3"/><i><font color="Green">E63</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>
 ;<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">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="E5:10_1_2_1_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E6:10_1_2_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font>
 <b>and </b><br/><a NAME="E7:10_1_2_1_3"/><i><font color="Green">E65</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">z'</font>
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T2" target="_self">Th2</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">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="E9:10_1_2_1_3"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>and </b><br/><a NAME="E10:10_1_2_1_3"/><i><font color="Green">E67</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="E11:10_1_2_1_3"/><i><font color="Green">E68</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">y'</font>
 <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>, , </i>;<br/><a NAME="E12:10_1_2_1_3"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x''</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#T3" target="_self">Th3</a>, , <a class="ref" href="arytm_0.html#T4" target="_self">Th4</a>, <a class="ref" href="arytm_2.html#T6">ARYTM_2:6</a></i>;<br/><a NAME="E13:10_1_2_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E14:10_1_2_1_3"/><b>then </b><i><font color="Green">E70</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <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/><a NAME="E15:10_1_2_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/><b>hence </b><a NAME="E16:10_1_2_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T5" target="_self">Th5</a>, , , , <a class="ref" href="arytm_1.html#T19">ARYTM_1:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:10_1_2_1_4"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E2:10_1_2_1_4"/><i><font color="Green">E72</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_1_2_1_4"/><i><font color="Green">E73</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_1_2_1_4"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E6:10_1_2_1_4"/><i><font color="Green">E77</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_1_2_1_4"/><i><font color="Green">E78</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">y'</font>
 <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>, , </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_1_2_1_4"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>and </b><br/><a NAME="E10:10_1_2_1_4"/><i><font color="Green">E80</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_1_2_1_4"/><i><font color="Green">E81</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">z'</font>
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T2" target="_self">Th2</a>, , </i>;<br/><font color="Maroon">y'</font> = 
<font color="Maroon">x'</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T4" target="_self">Th4</a>, </i>
<br/>.= 
<font color="Maroon">z'</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T3" target="_self">Th3</a>, <a class="ref" href="arytm_0.html#T5" target="_self">Th5</a>, , </i>
;<br/><b>hence </b><a NAME="E13:10_1_2_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_5"><b>suppose </b></a><b>that </b><a NAME="E1:10_1_2_1_5"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E2:10_1_2_1_5"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E3:10_1_2_1_5"/><i><font color="Green">E84</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>
 ;<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_1_2_1_5"/><i><font color="Green">E85</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_1_2_1_5"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 <b>and </b><br/><a NAME="E7:10_1_2_1_5"/><i><font color="Green">E87</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T1" target="_self">Th1</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_1_2_1_5"/><i><font color="Green">E88</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_1_2_1_5"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font>
 <b>and </b><br/><a NAME="E11:10_1_2_1_5"/><i><font color="Green">E90</font></i>: 
0 <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">x''</font>
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T2" target="_self">Th2</a>, , </i>;<br/><a NAME="E12:10_1_2_1_5"/>
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T3" target="_self">Th3</a>, <a class="ref" href="arytm_0.html#T5" target="_self">Th5</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>then </b><font color="Maroon">z'</font> = 
<font color="Maroon">x'</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">y'</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T4" target="_self">Th4</a>, </i>
;<br/><b>hence </b><a NAME="E14:10_1_2_1_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_6"><b>suppose </b></a><a NAME="E1:10_1_2_1_6"/>
( (  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> ) &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> ) )
 ;<br/><div class="add"><a NAME="E2:10_1_2_1_6"/><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> <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; 0 <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#K7">+</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>, </i>;<br/><b>hence </b><a NAME="E3:10_1_2_1_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_7"><b>suppose </b></a><a NAME="E1:10_1_2_1_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">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> ) &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> ) )
 ;<br/><div class="add"><a NAME="E2:10_1_2_1_7"/><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> <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; 0 <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#K7">+</a> <font color="Olive">z'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:10_1_2_1_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
