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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>



<b>assume </b><a NAME="E1:47"/><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> or  not 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><b>suppose </b></a><a NAME="E1:47_1_1"/><i><font color="Green">E32</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:47_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="extreal2.html#E1:47"><i><font color="Green">E31</font></i></a>, <a class="ref" href="extreal1.html#D3">EXTREAL1:def 3</a></i>;<br/><a NAME="E3:47_1_1"/>
 <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K3">-</a> <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E1:47_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="supinf_2.html#T16">SUPINF_2:16</a></i>;<br/><a NAME="E4:47_1_1"/><b>then </b>
 <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal2.html#E1:47_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="extreal1.html#T24">EXTREAL1:24</a></i>;<br/><a NAME="E5:47_1_1"/><b>then </b>
 <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="extreal2.html#E2:47_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E6:47_1_1"/>
(  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="extreal2.html#E1:47"><i><font color="Green">E31</font></i></a>, <a class="txt" href="extreal2.html#E1:47_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="extreal2.html#T47" target="_self">Th47</a>, <a class="ref" href="extreal1.html#D3">EXTREAL1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2"><b>suppose </b></a><a NAME="E1:47_1_2"/><i><font color="Green">E32</font></i>: 
not 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:47_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="extreal1.html#D3">EXTREAL1:def 3</a></i>;<br/><a NAME="E3:47_1_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal2.html#E1:47_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="extreal1.html#T25">EXTREAL1:25</a></i>;<br/><a NAME="E4:47_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="extreal2.html#E1:47"><i><font color="Green">E31</font></i></a>, <a class="txt" href="extreal2.html#E2:47_1_2"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E5:47_1_2"/>
(  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="extreal2.html#E1:47"><i><font color="Green">E31</font></i></a>, <a class="txt" href="extreal2.html#E1:47_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="extreal2.html#E2:47_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="extreal2.html#T47" target="_self">Th47</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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