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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"> ( <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"><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> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:13"/><span class="lab"><font color="Green" title="E11">A1</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="firebrick">::  thesis: </font></span></a><span class="hide"> <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> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <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></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_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="E1:13_1"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1"><span class="kw">suppose </span></a><a NAME="E1:13_1_1"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</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"><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> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <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></span><br/><div class="add"><a NAME="E2:13_1_1"/><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="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="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extreal1.html#D1" title="EXTREAL1:def.1">EXTREAL1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:13_1_1"/>
<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> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_1"><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="13_1_2"><span class="kw">suppose </span></a><a NAME="E1:13_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</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"><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> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <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></span><br/><div class="add"><a NAME="E2:13_1_2"/><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="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="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <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_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extreal1.html#D1" title="EXTREAL1:def.1">EXTREAL1:def 1</a>, <a class="ref" href="xxreal_3.html#T5" title="XXREAL_3:th.5">XXREAL_3:5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:13_1_2"/>
<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> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <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 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="13_1_3"><span class="kw">suppose </span></a><a NAME="E1:13_1_3"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</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"><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> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <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></span><br/><div class="add"><a NAME="E2:13_1_3"/><span class="lab"><font color="Green" title="E13">A4</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> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="extreal2.html#T4" target="_self" title="EXTREAL2:th.4">Th52</a></span>;<br/><a NAME="E3:13_1_3"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<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">by</span> <span class="lab"><a class="txt" href="#E1:13_1_3"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></span>;<br/><a NAME="E4:13_1_3"/>
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</a></span>;<br/><a NAME="E5:13_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A6</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_3"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:13_1_3"/><span class="kw">then </span>
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#D3" title="XXREAL_3:def.3">XXREAL_3:def 3</a></span>;<br/><a NAME="E7:13_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A7</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="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">by</span> <span class="lab"><a class="txt" href="#E3:13_1_3"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="extreal2.html#T11" target="_self" title="EXTREAL2:th.11">Th59</a></span>;<br/><a NAME="E8:13_1_3"/>
<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="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <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="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E5:13_1_3"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E3:13_1_3"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="extreal2.html#T14" target="_self" title="EXTREAL2:th.14">Th62</a></span>;<br/><span class="kw">hence </span><a NAME="E9:13_1_3"/>
<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> <a href="extreal1.html#K2" title="EXTREAL1:func.2">/</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1_3"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E7:13_1_3"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="xxreal_3.html#T88" title="XXREAL_3:th.88">XXREAL_3:88</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>

</div>
