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<b>let </b><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font></span><br/><b>let </b><font color="Maroon" title="c3">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">r</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font> implies <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:11"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E2:11"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font></span><br/>

<a NAME="E3:11"/><i><font color="Green" title="E10">A3</font></i>: 
( <font color="Maroon" title="c3">r</font> is <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> or <font color="Maroon" title="c3">r</font> is <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a> )
 
<b>by </b><i><a class="txt" href="#E1:11"><i><font color="Green" title="E8">A1</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1"/>
( <font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  or <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  or <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1"><b>suppose </b></a><a NAME="E1:11_1_1"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> being  <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:11_1_1"/><i><font color="Green" title="E12">A5</font></i>: 
( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )
 <b>and </b><br/><a NAME="E4:11_1_1"/><i><font color="Green" title="E13">A6</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_1_1"/>
not <font color="Maroon" title="c2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:11_1_1_1"/><b>then </b>
( <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  or <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><b>hence </b><a NAME="E3:11_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:11"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="#E2:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="#E1:11_1_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E6:11_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d</font> being <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">d</font> &amp; <font color="Maroon" title="c3">r</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">d</font> )
 <b>by </b><i><a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></i>;<br/><b>hence </b><a NAME="E7:11_1_1"/>
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="#E1:11"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="#E2:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="#E3:11_1_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="#E4:11_1_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_2"><b>suppose </b></a><a NAME="E1:11_1_2"/><i><font color="Green" title="E11">A7</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font></span><br/><div class="add"><a NAME="E2:11_1_2"/><b>then </b><i><font color="Green" title="E12">A8</font></i>: 
( <font color="Maroon" title="c3">r</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  or <font color="Maroon" title="c3">r</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  )
 <b>by </b><i><a class="txt" href="#E3:11"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></i>;<br/><b>assume </b><a NAME="E3:11_1_2"/>
<font color="Maroon" title="c1">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E4:11_1_2"/><b>then </b>
( <font color="Maroon" title="c2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  or <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  )
 <b>by </b><i><a class="txt" href="#E1:11_1_2"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><b>hence </b><a NAME="E5:11_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:11"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="#E2:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="#E1:11_1_2"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="#E2:11_1_2"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_3"><b>suppose </b></a><a NAME="E1:11_1_3"/><i><font color="Green" title="E11">A9</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font></span><br/><div class="add"><a NAME="E2:11_1_3"/><b>then </b><i><font color="Green" title="E12">A10</font></i>: 
( <font color="Maroon" title="c3">r</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  or <font color="Maroon" title="c3">r</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  )
 <b>by </b><i><a class="txt" href="#E3:11"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></i>;<br/><b>assume </b><a NAME="E3:11_1_3"/>
<font color="Maroon" title="c1">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E4:11_1_3"/><b>then </b>
( <font color="Maroon" title="c2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  or <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  )
 <b>by </b><i><a class="txt" href="#E1:11_1_3"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><b>hence </b><a NAME="E5:11_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:11"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="#E2:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="#E1:11_1_3"><i><font color="Green" title="E11">A9</font></i></a>, <a class="txt" href="#E2:11_1_3"><i><font color="Green" title="E12">A10</font></i></a>, <a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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