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<b>let </b><font color="Maroon">a</font> be  <a href="xxreal_0.html#V1" target="_self">ext-real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">b</font> be  <a href="xxreal_0.html#V1" target="_self">ext-real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1" target="_self">&lt;=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1" target="_self">&lt;=</a> <font color="Maroon">a</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">b</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">b</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">a</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">a</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">b</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">a</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K2" target="_self">-infty</a>  or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K1" target="_self">+infty</a>  ) &amp; ( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K2" target="_self">-infty</a>  or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K1" target="_self">+infty</a>  ) ) or ( (  not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) &amp; (  not <font color="Maroon">a</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">b</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">b</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">a</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">a</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">b</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="13_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:13_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E2:13_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">b</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">a'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">b'</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="E4:13_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 <b>and </b><br/><a NAME="E5:13_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 <b>and </b><br/><a NAME="E6:13_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">a'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">b'</font>
 <b>by </b><i>, , , </i>;<br/><b>consider </b><font color="Maroon">b''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">a''</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="E8:13_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b''</font>
 <b>and </b><br/><a NAME="E9:13_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a''</font>
 <b>and </b><br/><a NAME="E10:13_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">b''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">a''</font>
 <b>by </b><i>, , , </i>;<br/><b>thus </b><a NAME="E11:13_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D1" target="_self">Def1</a>, , <a class="ref" href="xxreal_0.html#D5" target="_self">Def5</a>, , , <a class="txt" href="xxreal_0.html#E1:13"><i><font color="Green">E19</font></i></a>, <a class="ref" href="arytm_1.html#T4">ARYTM_1:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_2"><b>suppose </b></a><a NAME="E1:13_1_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">b</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="E2:13_1_2"/>
( not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; not <font color="Maroon">a</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="E3:13_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_3"><b>suppose </b></a><a NAME="E1:13_1_3"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">a</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="E2:13_1_3"/>
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; not <font color="Maroon">b</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="E3:13_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:13_1_4"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</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="E2:13_1_4"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">b</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">a'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">b'</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="E4:13_1_4"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E5:13_1_4"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">b'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E6:13_1_4"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">b'</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">a'</font>
 <b>by </b><i>, , , </i>;<br/><b>consider </b><font color="Maroon">b''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">a''</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="E8:13_1_4"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">b''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E9:13_1_4"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">a''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E10:13_1_4"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a''</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">b''</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E11:13_1_4"/>
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a''</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b''</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#D1" target="_self">Def1</a>, , , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E12:13_1_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D5" target="_self">Def5</a>, , , <a class="txt" href="xxreal_0.html#E1:13"><i><font color="Green">E19</font></i></a>, <a class="ref" href="arytm_1.html#T4">ARYTM_1:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_5"><b>suppose </b></a><a NAME="E1:13_1_5"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K2" target="_self">-infty</a>  or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K1" target="_self">+infty</a>  ) &amp; ( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K2" target="_self">-infty</a>  or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K1" target="_self">+infty</a>  ) )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:13_1_5"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_6"><b>suppose </b></a><b>that </b><a NAME="E1:13_1_6"/><i><font color="Green">E40</font></i>: 
(  not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  )
 <b>and </b><br/><a NAME="E2:13_1_6"/><i><font color="Green">E41</font></i>: 
(  not <font color="Maroon">a</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">b</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="E3:13_1_6"/><i><font color="Green">E42</font></i>: 
(  not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">a</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="E4:13_1_6"/><i><font color="Green">E58</font></i>: 
(  not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or  not <font color="Maroon">b</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="E5:13_1_6"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K2" target="_self">-infty</a>  or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K1" target="_self">+infty</a>  )
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#D1" target="_self">Def1</a>, </i>;<br/><b>hence </b><a NAME="E6:13_1_6"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
