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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><b>that </b><br/><a NAME="E1:18"/><i><font color="Green">E15</font></i>: 
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>and </b><br/><a NAME="E3:18"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="extreal2.html#E1:18"><i><font color="Green">E15</font></i></a>, <a class="txt" href="extreal2.html#E2:18"><i><font color="Green">E16</font></i></a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<a NAME="E5:18"/><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="extreal1.html#K3">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_1_1_1"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="extreal2.html#E3:18"><i><font color="Green">E17</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_1_1_1_1"/><i><font color="Green">E20</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E1:18_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:18_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="extreal1.html#K3">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="extreal2.html#E2:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="extreal2.html#E1:18_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="extreal2.html#E1:18_1_1_1_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal2.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_1_1_1_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:18_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E1:18_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:18_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="extreal1.html#K3">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="extreal2.html#E1:18"><i><font color="Green">E15</font></i></a>, <a class="txt" href="extreal2.html#E1:18_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="extreal2.html#E1:18_1_1_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal2.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:18_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="extreal1.html#K3">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_2_1_1"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="extreal2.html#E3:18"><i><font color="Green">E17</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_2_1_1_1"/><i><font color="Green">E20</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E1:18_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:18_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="extreal1.html#K3">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="extreal2.html#E2:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="extreal2.html#E1:18_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="extreal2.html#E1:18_1_1_2_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal2.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_2_1_1_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:18_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E1:18_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:18_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="extreal1.html#K3">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="extreal2.html#E1:18"><i><font color="Green">E15</font></i></a>, <a class="txt" href="extreal2.html#E1:18_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="extreal2.html#E1:18_1_1_2_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal2.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:18_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="extreal1.html#K3">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_3"><b>suppose </b></a><a NAME="E1:18_1_1_3"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><a NAME="E3:18_1_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="extreal1.html#K3">/</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="extreal2.html#E3:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="extreal1.html#T32">EXTREAL1:32</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="extreal2.html#E3:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
;<br/><b>hence </b><a NAME="E5:18_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="extreal1.html#K3">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:18"/>
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K3">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K10">1.</a> 
 
<b>by </b><i><a class="txt" href="extreal2.html#E1:18"><i><font color="Green">E15</font></i></a>, <a class="txt" href="extreal2.html#E2:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="extreal2.html#E3:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="extreal1.html#T34">EXTREAL1:34</a>, <a class="ref" href="mesfunc1.html#D8">MESFUNC1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E7:18"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="extreal1.html#K3">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K3">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="extreal2.html#E5:18"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal2.html#T4" target="_self">Th4</a></i>;<br/>


</div>
