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<span class="kw">let </span><font color="Maroon" title="c1">x</font> be   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:15_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> or <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_1"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span></span><br/><div class="add"><span class="kw">then </span><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> = 
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="extreal1.html#T4" title="EXTREAL1:th.4">EXTREAL1:4</a></span>
<br/>.= 
<font color="Maroon" title="c1">x</font>

;<br/><span class="kw">hence </span><a NAME="E3:15_1_1"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="extreal1.html#T3" title="EXTREAL1:th.3">EXTREAL1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_2"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span></span><br/><div class="add"><a NAME="E2:15_1_2"/><span class="kw">then </span>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extreal1.html#T4" title="EXTREAL1:th.4">EXTREAL1:4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:15_1_2"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_2"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="extreal1.html#T3" title="EXTREAL1:th.3">EXTREAL1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_3"><span class="kw">suppose </span></a><a NAME="E1:15_1_3"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_3"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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