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<b>let </b><font color="Maroon" title="c1">f</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"> <span class="p1">(<span class="default"><a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_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="6_1_1"><b>suppose </b></a><a NAME="E1:6_1_1"/><i><font color="Green" title="E3">A1</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"> <span class="p1">(<span class="default"><a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c2">g</font> = <font color="Maroon" title="c1">f</font> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/><b>consider </b><font color="Maroon" title="c3">a</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="E4:6_1_1"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <b>and </b><br/><a NAME="E5:6_1_1"/><i><font color="Green" title="E5">A3</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
 <b>by </b><i><a class="txt" href="#E1:6_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="xxreal_3.html#D4" title="XXREAL_3:def.4">XXREAL_3:def 4</a></i>;<br/><a NAME="E6:6_1_1"/><i><font color="Green" title="E6">A4</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T224" title="XCMPLX_1:th.224">XCMPLX_1:224</a></i>;<br/><a NAME="E7:6_1_1"/><i><font color="Green" title="E7">A5</font></i>: 
 <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c2">g</font>
 ;<br/><a NAME="E8:6_1_1"/>
 ex <font color="Olive" title="b1">m</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/>(  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> &amp;  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">a</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">a</font> &amp;  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  )</span><br/>

<b>thus </b><a NAME="E1:6_1_1_1"/>
 <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="#E4:6_1_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="#E7:6_1_1"><i><font color="Green" title="E7">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span><br/>

<b>thus </b><a NAME="E2:6_1_1_1"/>
 <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
 <b>by </b><i><a class="txt" href="#E5:6_1_1"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="#E7:6_1_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="#E6:6_1_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="xxreal_3.html#D3" title="XXREAL_3:def.3">XXREAL_3:def 3</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><b>hence </b><a NAME="E9:6_1_1"/>
<span class="p1">(<span class="default"><a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E7:6_1_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="xxreal_3.html#D4" title="XXREAL_3:def.4">XXREAL_3:def 4</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="6_1_2"><b>suppose </b></a><a NAME="E1:6_1_2"/><i><font color="Green" title="E3">A6</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"> <span class="p1">(<span class="default"><a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  = 
<a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> 
<b>by </b><i><a class="ref" href="xxreal_3.html#D3" title="XXREAL_3:def.3">XXREAL_3:def 3</a></i>
<br/>.= 
 <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

<br/>.= 
 <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E1:6_1_2"><i><font color="Green" title="E3">A6</font></i></a></i>
;<br/> <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="6_1_3"><b>suppose </b></a><a NAME="E1:6_1_3"/><i><font color="Green" title="E3">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#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"> <span class="p1">(<span class="default"><a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  = 
<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> 
<b>by </b><i><a class="ref" href="xxreal_3.html#D3" title="XXREAL_3:def.3">XXREAL_3:def 3</a></i>
<br/>.= 
 <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

<br/>.= 
 <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E1:6_1_3"><i><font color="Green" title="E3">A7</font></i></a></i>
;<br/> <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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