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<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:51"/><i><font color="Green" title="E35">A1</font></i>: 
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font>
 <b>and </b><br/><a NAME="E2:51"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>and </b><br/><a NAME="E3:51"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c2">a</font> = <font color="Maroon" title="c1">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="extreal2.html#E1:51"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="extreal2.html#E2:51"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/>
<a NAME="E5:51"/><i><font color="Green" title="E38">A4</font></i>: 
<a href="mesfunc1.html#K10" title="MESFUNC1:func.10">1.</a>  <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="extreal2.html#E3:51"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="extreal1.html#T32" title="EXTREAL1:th.32">EXTREAL1:32</a>, <a class="ref" href="mesfunc1.html#D8" title="MESFUNC1:def.8">MESFUNC1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> or  not 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1"><b>suppose </b></a><a NAME="E1:51_1_1"/><i><font color="Green" title="E39">A5</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 ;<br/><div class="add"><a NAME="E2:51_1_1"/><b>then </b><i><font color="Green" title="E40">A6</font></i>: 
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="extreal1.html#D3" title="EXTREAL1:def.3">EXTREAL1:def 3</a></i>;<br/><a NAME="E3:51_1_1"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="extreal2.html#E3:51"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="extreal2.html#E1:51_1_1"><i><font color="Green" title="E39">A5</font></i></a></i>;<br/><a NAME="E4:51_1_1"/><b>then </b><i><font color="Green" title="E41">A7</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">a</font>
 ;<br/><a NAME="E5:51_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/><a NAME="E6:51_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">1.</a>  <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="extreal2.html#E5:51"><i><font color="Green" title="E38">A4</font></i></a></i>;<br/><a NAME="E7:51_1_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K10" title="MESFUNC1:func.10">1.</a>  <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="extreal2.html#E5:51"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="extreal1.html#T36" title="EXTREAL1:th.36">EXTREAL1:36</a></i>;<br/><a NAME="E8:51_1_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K10" title="MESFUNC1:func.10">1.</a>  <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal2.html#E2:51_1_1"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>;<br/><b>hence </b><a NAME="E9:51_1_1"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K10" title="MESFUNC1:func.10">1.</a>  <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">1.</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E4:51_1_1"><i><font color="Green" title="E41">A7</font></i></a>, <a class="ref" href="mesfunc1.html#D8" title="MESFUNC1:def.8">MESFUNC1:def 8</a>, <a class="ref" href="xcmplx_1.html#T107" title="XCMPLX_1:th.107">XCMPLX_1:107</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_2"><b>suppose </b></a><a NAME="E1:51_1_2"/><i><font color="Green" title="E39">A8</font></i>: 
not 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E40">A9</font></i>: <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> = 
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="ref" href="extreal1.html#D3" title="EXTREAL1:def.3">EXTREAL1:def 3</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="supinf_2.html#T3" title="SUPINF_2:th.3">SUPINF_2:3</a></i>
;<br/><a NAME="E3:51_1_2"/><i><font color="Green" title="E41">A10</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal2.html#E1:51_1_2"><i><font color="Green" title="E39">A8</font></i></a></i>;<br/><a NAME="E4:51_1_2"/><b>then </b>
1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T149" title="REAL_2:th.149">REAL_2:149</a></i>;<br/><a NAME="E5:51_1_2"/><b>then </b>
<a href="mesfunc1.html#K10" title="MESFUNC1:func.10">1.</a>  <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal2.html#E5:51"><i><font color="Green" title="E38">A4</font></i></a></i>;<br/><b>then </b><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K10" title="MESFUNC1:func.10">1.</a>  <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> = 
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K10" title="MESFUNC1:func.10">1.</a>  <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c1">x</font></span>)</span>
<b>by </b><i><a class="ref" href="extreal1.html#T37" title="EXTREAL1:th.37">EXTREAL1:37</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">a</font></span>)</span>
<b>by </b><i><a class="txt" href="extreal2.html#E5:51"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="supinf_2.html#T3" title="SUPINF_2:th.3">SUPINF_2:3</a></i>
;<br/><b>then </b><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K10" title="MESFUNC1:func.10">1.</a>  <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="extreal2.html#E2:51_1_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">a</font></span>)</span>

;<br/><b>hence </b><a NAME="E8:51_1_2"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K10" title="MESFUNC1:func.10">1.</a>  <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K10" title="MESFUNC1:func.10">1.</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E3:51_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="mesfunc1.html#D8" title="MESFUNC1:def.8">MESFUNC1:def 8</a>, <a class="ref" href="xcmplx_1.html#T107" title="XCMPLX_1:th.107">XCMPLX_1:107</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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