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



<b>assume </b><a NAME="E1:61"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">x</font> is   <a href="real_1.html#NM1">Real</a> or <font color="Maroon">y</font> is   <a href="real_1.html#NM1">Real</a> )
 ;<br/>

<a NAME="E2:61"/><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K4">-</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 
<b>by </b><i>, <a class="ref" href="extreal2.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E3:61"/><i><font color="Green">E20</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K4">-</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 
<b>by </b><i>, <a class="ref" href="extreal2.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E4:61"/>
<font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:61"/><b>then </b><i><font color="Green">E24</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 
<b>by </b><i><a class="ref" href="extreal2.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E6:61"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K4">-</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K4">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:61"/><b>then </b>
 <a href="supinf_2.html#K3">-</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K4">-</a> <span class="p3"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="supinf_2.html#T16">SUPINF_2:16</a></i>;<br/>
<b>hence </b><a NAME="E8:61"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K4">-</a> <span class="p3"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i>, <a class="txt" href="extreal2.html#E21"><i><font color="Green">Lemma32</font></i></a></i>;<br/>


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