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<b>let </b><font color="Maroon">x</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:20"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:20"/><i><font color="Green">E18</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x2</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:20"/><i><font color="Green">E19</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x1</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> =  <a href="supinf_1.html#K7">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="domain_1.html#K7">}</a></span> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1"/>
( <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> or <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><b>suppose </b></a><a NAME="E1:20_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font>
 ;<br/><div class="add"><a NAME="E2:20_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T96">SUPINF_1:96</a></i>;<br/><b>thus </b><a NAME="E3:20_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="supinf_1.html#T96">SUPINF_1:96</a></i>;<br/><a NAME="E4:20_1_1"/>
<font color="Maroon">x1</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i>, </i>;<br/><a NAME="E5:20_1_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x1</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E6:20_1_1"/>
<font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2"><b>suppose </b></a><a NAME="E1:20_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font>
 ;<br/><div class="add"><a NAME="E2:20_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T96">SUPINF_1:96</a></i>;<br/><b>thus </b><a NAME="E3:20_1_2"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="supinf_1.html#T96">SUPINF_1:96</a></i>;<br/><a NAME="E4:20_1_2"/>
( <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x1</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E5:20_1_2"/>
<font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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