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

<b>assume </b><a NAME="E1:16"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>reconsider </b><font color="Maroon">y'</font> = 0 as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  <b>by </b><i><a class="ref" href="arytm_2.html#T21">ARYTM_2:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1"/>
( <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">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><b>suppose </b></a><a NAME="E1:16_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  ;<br/><a NAME="E3:16_1_1"/>
<font color="Maroon">x'</font> <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_2.html#D8">ARYTM_2:def 8</a></i>;<br/><b>hence </b><a NAME="E4:16_1_1"/>
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2"><b>suppose </b></a><a NAME="E1:16_1_2"/><i><font color="Green">E16</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/><div class="add"><a NAME="E2:16_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="arytm_2.html#K2">REAL+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="arytm_3.html#K13">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="numbers.html#D1">NUMBERS:def 1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:16_1_2"/><b>then </b><i><font color="Green">E23</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"><a href="arytm_3.html#K13">{}</a> </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#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:16_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="arytm_3.html#K13">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E6:16_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E7:16_1_2"/><i><font color="Green">E36</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"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  <b>by </b><i/>;<br/><a NAME="E9:16_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E10:16_1_2"/><b>then </b>
<font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:16_1_2"><i><font color="Green">E23</font></i></a>, </i>;<br/><a NAME="E11:16_1_2"/><b>then </b>
<font color="Maroon">x</font> <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="txt" href="arytm_0.html#E3:16_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="arytm_0.html#E5:16_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="arytm_1.html#T19">ARYTM_1:19</a></i>;<br/><b>hence </b><a NAME="E12:16_1_2"/>
 <a href="arytm_0.html#K1" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T5" target="_self">Th5</a>, <a class="txt" href="arytm_0.html#E3:16_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="arytm_0.html#E5:16_1_2"><i><font color="Green">E24</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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