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

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1" target="_self">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1" target="_self">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">t</font> be  <a href="xreal_0.html#V1" target="_self">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>assume </b><a NAME="E1:10"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">r</font>, <font color="Maroon">y1</font> = <font color="Maroon">s</font>, <font color="Maroon">z1</font> = <font color="Maroon">t</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/>
<a NAME="E3:10"/>
for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <br/> for <font color="Olive">r</font> being <a href="xreal_0.html#V1" target="_self">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> holds <br/> <a href="arytm_0.html#K1">+</a> <font color="Olive">x'</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_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> ;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1" target="_self">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/>

<b>consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:10_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E4:10_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E5:10_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D4">XCMPLX_0:def 4</a></i>;<br/>
<a NAME="E6:10_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:10_1"/>
( <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:10_1"/><b>then </b>
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>;<br/>
<b>hence </b><a NAME="E9:10_1"/>
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x'</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 <b>by </b><i>, , , <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:10"/><b>then </b><i><font color="Green">E33</font></i>: 
(  <a href="arytm_0.html#K1">+</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> &amp;  <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> )
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_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"/>
( ( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) or ( <font color="Maroon">r</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">s</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) or ( <font color="Maroon">r</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">s</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">t</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) or ( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">t</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">r</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">s</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">t</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">r</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">s</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">t</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>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1"><b>suppose </b></a><b>that </b><a NAME="E1:10_2_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E2:10_2_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E3:10_2_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<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">s''</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_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>and </b><br/><a NAME="E6:10_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s''</font>
 <b>and </b><br/><a NAME="E7:10_2_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">s''</font>
 <b>by </b><i>, , , </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">t'</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_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E10:10_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t'</font>
 <b>and </b><br/><a NAME="E11:10_2_1"/><i><font color="Green">E43</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">t'</font>
 <b>by </b><i>, , <a class="ref" href="arytm_0.html#D2">ARYTM_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon">s'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">t''</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="E13:10_2_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s'</font>
 <b>and </b><br/><a NAME="E14:10_2_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t''</font>
 <b>and </b><br/><a NAME="E15:10_2_1"/><i><font color="Green">E58</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">t''</font>
 <b>by </b><i>, , <a class="ref" href="arytm_0.html#D2">ARYTM_0:def 2</a></i>;<br/><a NAME="E16:10_2_1"/>
<font color="Maroon">x'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">s'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">t''</font>
 <b>by </b><i>, , , , , <a class="ref" href="xreal_0.html#D1" target="_self">Def1</a>, <a class="txt" href="xreal_0.html#E1:10"><i><font color="Green">E20</font></i></a>, <a class="ref" href="arytm_1.html#T7">ARYTM_1:7</a></i>;<br/><b>hence </b><a NAME="E17:10_2_1"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2"><b>suppose </b></a><b>that </b><a NAME="E1:10_2_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">r</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"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E3:10_2_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<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">t'</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"/>
<font color="Maroon">r</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"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t'</font>
 <b>and </b><br/><a NAME="E7:10_2_2"/><i><font color="Green">E68</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">x'</font>
 <b>by </b><i>, , <a class="ref" href="arytm_0.html#D2">ARYTM_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon">s'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">t''</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"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s'</font>
 <b>and </b><br/><a NAME="E10:10_2_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t''</font>
 <b>and </b><br/><a NAME="E11:10_2_2"/><i><font color="Green">E70</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">t''</font>
 <b>by </b><i>, , <a class="ref" href="arytm_0.html#D2">ARYTM_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_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"/>
( <font color="Maroon">x'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font> or  not <font color="Maroon">x'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:10_2_2_1_1_1"/>
<font color="Maroon">x'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font>
 ;<br/><div class="add"><a NAME="E2:10_2_2_1_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">t'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:10_2_2_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">t'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">x'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#T11">ARYTM_1:11</a></i>;<br/><a NAME="E4:10_2_2_1_1_1"/>
<font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">s'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">t''</font>
 <b>by </b><i>, , <a class="ref" href="arytm_2.html#T20">ARYTM_2:20</a></i>;<br/><a NAME="E5:10_2_2_1_1_1"/><b>then </b>
<font color="Maroon">t'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">x'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">s'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">t''</font>
 <b>by </b><i>, <a class="ref" href="arytm_1.html#T3">ARYTM_1:3</a></i>;<br/><b>hence </b><a NAME="E6:10_2_2_1_1_1"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:10_2_2_1_1_2"/>
not <font color="Maroon">x'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font>
 ;<br/><div class="add"><a NAME="E2:10_2_2_1_1_2"/><b>then </b>
<font color="Maroon">t'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">x'</font> <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="Maroon">x'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">t'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:10_2_2_1_1_2"/><b>then </b>
<font color="Maroon">t'</font> <a href="arytm_1.html#K2">-</a> <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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:10_2_2_1_1_2"/><b>then </b>
( not <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; not <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</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#T5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E5:10_2_2_1_1_2"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:10_2_2"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_3"><b>suppose </b></a><b>that </b><a NAME="E1:10_2_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">r</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_3"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">s</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_3"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<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">s''</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_3"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">r</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_3"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">s''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E7:10_2_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">s''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">x''</font>
 <b>by </b><i>, , , </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">t'</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_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">r</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_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t'</font>
 <b>and </b><br/><a NAME="E11:10_2_3"/><i><font color="Green">E81</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">x'</font>
 <b>by </b><i>, , <a class="ref" href="arytm_0.html#D2">ARYTM_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon">s'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">t''</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="E13:10_2_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">s'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E14:10_2_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t''</font>
 <b>and </b><br/><a NAME="E15:10_2_3"/><i><font color="Green">E84</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t''</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">s'</font>
 <b>by </b><i>, , <a class="ref" href="arytm_0.html#D2">ARYTM_0:def 2</a></i>;<br/><a NAME="E16:10_2_3"/><i><font color="Green">E85</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="E17:10_2_3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s''</font>
 <b>by </b><i>, <a class="ref" href="xreal_0.html#D1" target="_self">Def1</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_3_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_3_1_1"/>
( <font color="Maroon">x'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font> or  not <font color="Maroon">x'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:10_2_3_1_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">x'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font>
 ;<br/><div class="add"><a NAME="E2:10_2_3_1_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">t'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:10_2_3_1_1_1"/>
<font color="Maroon">s'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font>
 <b>by </b><i>, , , <a class="txt" href="xreal_0.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="arytm_1.html#T3">ARYTM_1:3</a></i>;<br/><a NAME="E4:10_2_3_1_1_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">t'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">s'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>;<br/><a NAME="E5:10_2_3_1_1_1"/>
<font color="Maroon">t'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">x'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t''</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">s'</font>
 <b>by </b><i>, , <a class="txt" href="xreal_0.html#E1:10"><i><font color="Green">E20</font></i></a>, , , <a class="ref" href="arytm_1.html#T16">ARYTM_1:16</a></i>;<br/><b>hence </b><a NAME="E6:10_2_3_1_1_1"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 <b>by </b><i>, , , <a class="txt" href="xreal_0.html#E1:10"><i><font color="Green">E20</font></i></a>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:10_2_3_1_1_2"/>
not <font color="Maroon">x'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font>
 ;<br/><div class="add"><a NAME="E2:10_2_3_1_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font> <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="Maroon">x'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">t'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:10_2_3_1_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_3_1_1_2_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_3_1_1_2_1_1"/>
( <font color="Maroon">s'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font> or  not <font color="Maroon">s'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:10_2_3_1_1_2_1_1_1"/>
<font color="Maroon">s'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font>
 ;<br/><div class="add"><a NAME="E2:10_2_3_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">t'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">s'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:10_2_3_1_1_2_1_1_1"/><b>then </b>
( not  <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; not  <a href="arytm_0.html#K1">+</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</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="txt" href="xreal_0.html#E1:10"><i><font color="Green">E20</font></i></a>, , , <a class="ref" href="arytm_0.html#T5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E4:10_2_3_1_1_2_1_1_1"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:10_2_3_1_1_2_1_1_2"/>
not <font color="Maroon">s'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font>
 ;<br/><div class="add"><a NAME="E2:10_2_3_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</font> <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="Maroon">s'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">t'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="txt" href="xreal_0.html#E1:10"><i><font color="Green">E20</font></i></a>, , <a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:10_2_3_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:10_2_3_1_1_2_1_1_2"/>
<font color="Maroon">s'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">x'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">t'</font>
 <b>by </b><i>, , , <a class="ref" href="arytm_1.html#T17">ARYTM_1:17</a></i>;<br/><b>hence </b><a NAME="E5:10_2_3_1_1_2_1_1_2"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="txt" href="xreal_0.html#E2"><i><font color="Green">Lemma24</font></i></a>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:10_2_3_1_1_2"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E19:10_2_3"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_4"><b>suppose </b></a><b>that </b><a NAME="E1:10_2_4"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E2:10_2_4"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E3:10_2_4"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">t</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">s''</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_4"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>and </b><br/><a NAME="E6:10_2_4"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s''</font>
 <b>and </b><br/><a NAME="E7:10_2_4"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">x''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">s''</font>
 <b>by </b><i>, , , </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">t'</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_4"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E10:10_2_4"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">t'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E11:10_2_4"/><i><font color="Green">E102</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">t'</font>
 <b>by </b><i>, , <a class="ref" href="arytm_0.html#D2">ARYTM_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon">s'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">t''</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="E13:10_2_4"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s'</font>
 <b>and </b><br/><a NAME="E14:10_2_4"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">t''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E15:10_2_4"/><i><font color="Green">E105</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">t''</font>
 <b>by </b><i>, , <a class="ref" href="arytm_0.html#D2">ARYTM_0:def 2</a></i>;<br/><a NAME="E16:10_2_4"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t''</font>
 <b>by </b><i>, <a class="txt" href="xreal_0.html#E1:10"><i><font color="Green">E20</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_4_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_4_1_1"/>
( <font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">x'</font> or  not <font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">x'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:10_2_4_1_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">x'</font>
 ;<br/><div class="add"><a NAME="E2:10_2_4_1_1_1"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">x'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">t'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:10_2_4_1_1_1"/>
<font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">s'</font>
 <b>by </b><i>, , , , <a class="ref" href="xreal_0.html#D1" target="_self">Def1</a>, , <a class="ref" href="arytm_1.html#T3">ARYTM_1:3</a></i>;<br/><a NAME="E4:10_2_4_1_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">s'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">t'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>;<br/><a NAME="E5:10_2_4_1_1_1"/>
<font color="Maroon">x'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">s'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">t''</font>
 <b>by </b><i>, , , , <a class="ref" href="xreal_0.html#D1" target="_self">Def1</a>, , <a class="ref" href="arytm_1.html#T17">ARYTM_1:17</a></i>;<br/><b>hence </b><a NAME="E6:10_2_4_1_1_1"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 <b>by </b><i>, , , , <a class="txt" href="xreal_0.html#E2"><i><font color="Green">Lemma24</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:10_2_4_1_1_2"/>
not <font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">x'</font>
 ;<br/><div class="add"><a NAME="E2:10_2_4_1_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font> <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="Maroon">t'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">x'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:10_2_4_1_1_2"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_4_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_4_1_1_2_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_4_1_1_2_1_1"/>
( <font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">s'</font> or  not <font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">s'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_4_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:10_2_4_1_1_2_1_1_1"/>
<font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">s'</font>
 ;<br/><div class="add"><a NAME="E2:10_2_4_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">s'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">t'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:10_2_4_1_1_2_1_1_1"/><b>then </b>
( not  <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; not  <a href="arytm_0.html#K1">+</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</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="txt" href="xreal_0.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="arytm_0.html#T5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E4:10_2_4_1_1_2_1_1_1"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_4_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:10_2_4_1_1_2_1_1_2"/>
not <font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">s'</font>
 ;<br/><div class="add"><a NAME="E2:10_2_4_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</font> <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="Maroon">t'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">s'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:10_2_4_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:10_2_4_1_1_2_1_1_2"/>
<font color="Maroon">t'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">s'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">x'</font>
 <b>by </b><i>, , , , <a class="ref" href="xreal_0.html#D1" target="_self">Def1</a>, <a class="ref" href="arytm_1.html#T16">ARYTM_1:16</a></i>;<br/><b>hence </b><a NAME="E5:10_2_4_1_1_2_1_1_2"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 <b>by </b><i>, , <a class="txt" href="xreal_0.html#E2"><i><font color="Green">Lemma24</font></i></a>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:10_2_4_1_1_2"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E18:10_2_4"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_5"><b>suppose </b></a><b>that </b><a NAME="E1:10_2_5"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">r</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_5"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E3:10_2_5"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">t</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="E4:10_2_5"/>
( not <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  )
 <b>by </b><i>, , <a class="ref" href="arytm_0.html#T5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>then 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">t'</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="E6:10_2_5"/>
<font color="Maroon">r</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="E7:10_2_5"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">t'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E8:10_2_5"/><i><font color="Green">E118</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font> <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="Maroon">x'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">t'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="arytm_0.html#D2">ARYTM_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon">s'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">t''</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="E10:10_2_5"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s'</font>
 <b>and </b><br/><a NAME="E11:10_2_5"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">t''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E12:10_2_5"/><i><font color="Green">E120</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">t''</font>
 <b>by </b><i>, , <a class="ref" href="arytm_0.html#D2">ARYTM_0:def 2</a></i>;<br/><a NAME="E13:10_2_5"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t''</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E14:10_2_5"/><i><font color="Green">E122</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_5_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_5_1_1"/>
( <font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">s'</font> or  not <font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">s'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_5_1_1_1"><b>suppose </b></a><a NAME="E1:10_2_5_1_1_1"/>
<font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">s'</font>
 ;<br/><div class="add"><a NAME="E2:10_2_5_1_1_1"/><b>then </b>
<font color="Maroon">s'</font> <a href="arytm_1.html#K2">-</a> <font color="Maroon">t''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">t''</font>
 <b>by </b><i>, <a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:10_2_5_1_1_1"/><b>then </b>
( not  <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; not  <a href="arytm_0.html#K1">+</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</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#T5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E4:10_2_5_1_1_1"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_5_1_1_2"><b>suppose </b></a><a NAME="E1:10_2_5_1_1_2"/>
not <font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">s'</font>
 ;<br/><div class="add"><a NAME="E2:10_2_5_1_1_2"/><b>then </b><i><font color="Green">E123</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</font> <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="Maroon">t'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">s'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="ref" href="arytm_1.html#D2">ARYTM_1:def 2</a></i>;<br/><a NAME="E3:10_2_5_1_1_2"/><b>then </b><i><font color="Green">E124</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:10_2_5_1_1_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">t'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">s'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#T11">ARYTM_1:11</a></i>;<br/><a NAME="E5:10_2_5_1_1_2"/>
<font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T20">ARYTM_2:20</a></i>;<br/><a NAME="E6:10_2_5_1_1_2"/><b>then </b>
<font color="Maroon">t'</font> <a href="arytm_1.html#K1">-'</a> <font color="Maroon">s'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">t'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">x'</font>
 <b>by </b><i>, <a class="ref" href="arytm_1.html#T3">ARYTM_1:3</a></i>;<br/><b>hence </b><a NAME="E7:10_2_5_1_1_2"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 <b>by </b><i>, , , <a class="ref" href="xreal_0.html#D1" target="_self">Def1</a>, <a class="txt" href="xreal_0.html#E1:10"><i><font color="Green">E20</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E16:10_2_5"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_6"><b>suppose </b></a><b>that </b><a NAME="E1:10_2_6"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">r</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_6"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">s</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_6"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">t</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">s''</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_6"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">r</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_6"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">s''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E7:10_2_6"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">s''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">x''</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E8:10_2_6"/>
( not <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  )
 <b>by </b><i>, , <a class="ref" href="arytm_0.html#T5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>then 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">t'</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="E10:10_2_6"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">r</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="E11:10_2_6"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">t'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E12:10_2_6"/><i><font color="Green">E134</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font> <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="Maroon">x'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">t'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="arytm_0.html#D2">ARYTM_0:def 2</a></i>;<br/><a NAME="E13:10_2_6"/>
( not <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  )
 <b>by </b><i>, , <a class="ref" href="arytm_0.html#T5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>then consider </b><font color="Maroon">s'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">t''</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="E15:10_2_6"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">s'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E16:10_2_6"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">t''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E17:10_2_6"/><i><font color="Green">E137</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</font> <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="Maroon">s'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">t''</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="arytm_0.html#D2">ARYTM_0:def 2</a></i>;<br/><a NAME="E18:10_2_6"/><i><font color="Green">E138</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="E19:10_2_6"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s''</font>
 <b>by </b><i>, <a class="ref" href="xreal_0.html#D1" target="_self">Def1</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E20:10_2_6"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t''</font>
 <b>by </b><i>, <a class="txt" href="xreal_0.html#E1:10"><i><font color="Green">E20</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E21:10_2_6"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">s'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">t'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">x'</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">t''</font>
 <b>by </b><i>, , , <a class="ref" href="arytm_1.html#T7">ARYTM_1:7</a></i>;<br/><a NAME="E22:10_2_6"/><i><font color="Green">E142</font></i>: 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E23:10_2_6"/>
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E24:10_2_6"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">t</font>
 <b>by </b><i>, , , <a class="txt" href="xreal_0.html#E2"><i><font color="Green">Lemma24</font></i></a>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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