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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="Red">::  thesis: </font></span></a><span class="hide"> (  <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> &amp; <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>  &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <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="extreal1.html#K1" title="EXTREAL1:func.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="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a>  <a href="extreal1.html#K2" title="EXTREAL1: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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:59"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
(  <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> &amp; <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>  )
 <span class="kw">and </span><br/><a NAME="E2:59"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="Red">::  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="extreal1.html#K1" title="EXTREAL1:func.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="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a>  <a href="extreal1.html#K2" title="EXTREAL1: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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> </span><br/>

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></span>;<br/>
<a NAME="E4:59"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
<a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a>  <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><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></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_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="E1:59_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> or  not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="59_1_1"><span class="kw">suppose </span></a><a NAME="E1:59_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="extreal1.html#K1" title="EXTREAL1:func.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="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a>  <a href="extreal1.html#K2" title="EXTREAL1: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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> </span><br/><div class="add"><a NAME="E2:59_1_1"/><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> <font color="Maroon" title="c2">a</font> &amp; <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a>  <a href="extreal1.html#K2" title="EXTREAL1: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="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="extreal1.html#D3" title="EXTREAL1:def.3">EXTREAL1:def 3</a></span>;<br/><a NAME="E3:59_1_1"/><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="extreal1.html#K1" title="EXTREAL1:func.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="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a>  <a href="extreal1.html#K2" title="EXTREAL1: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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:59_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="extreal1.html#K1" title="EXTREAL1:func.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="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a>  <a href="extreal1.html#K2" title="EXTREAL1: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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E12">A2</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="59_1_2"><span class="kw">suppose </span></a><a NAME="E1:59_1_2"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="extreal1.html#K1" title="EXTREAL1:func.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="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a>  <a href="extreal1.html#K2" title="EXTREAL1: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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> </span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A5</font></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="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c1">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="extreal1.html#D3" title="EXTREAL1:def.3">EXTREAL1:def 3</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T3" title="SUPINF_2:th.3">SUPINF_2:3</a></span>
;<br/><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a>  <a href="extreal1.html#K2" title="EXTREAL1: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#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a>  <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <font color="Maroon" title="c1">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E1:59_1_2"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="extreal1.html#T37" title="EXTREAL1:th.37">EXTREAL1:37</a>, <a class="ref" href="xreal_1.html#T144" title="XREAL_1:th.144">XREAL_1:144</a></span>
<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#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="supinf_2.html#T3" title="SUPINF_2:th.3">SUPINF_2:3</a></span>
;<br/><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="extreal1.html#K1" title="EXTREAL1:func.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="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a>  <a href="extreal1.html#K2" title="EXTREAL1: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="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#K8" title="REAL_1:func.8">*</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#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1_2"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">a</font></span>)</span>

;<br/><span class="kw">hence </span><a NAME="E5:59_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="extreal1.html#K1" title="EXTREAL1:func.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="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a>  <a href="extreal1.html#K2" title="EXTREAL1: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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_2"><span class="lab"><font color="Green" title="E14">A4</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
