<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</a></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="157_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="E1:157_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><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="157_1_1"><span class="kw">suppose </span></a><a NAME="E1:157_1_1"/><span class="lab"><font color="Green" title="E85">A1</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</a></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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/><a NAME="E3:157_1_1"/><span class="lab"><font color="Green" title="E86">A2</font></span>: 
 <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_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c2">g</font>
 ;<br/><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E5:157_1_1"/><span class="lab"><font color="Green" title="E87">A3</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">and </span><br/><a NAME="E6:157_1_1"/><span class="lab"><font color="Green" title="E88">A4</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:157_1_1"><span class="lab"><font color="Green" title="E85">A1</font></span></a>, <a class="ref" href="xxreal_3.html#D6" target="_self" title="XXREAL_3:def.6">Def6</a></span>;<br/><a NAME="E7:157_1_1"/><span class="lab"><font color="Green" title="E89">A5</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T222" title="XCMPLX_1:th.222">XCMPLX_1:222</a></span>;<br/><a NAME="E8:157_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#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">m</font> &amp;  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</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="157_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
 <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">a</font> &amp;  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</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/>

<span class="kw">thus </span><a NAME="E1:157_1_1_1"/>
 <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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:157_1_1"><span class="lab"><font color="Green" title="E87">A3</font></span></a>, <a class="txt" href="#E3:157_1_1"><span class="lab"><font color="Green" title="E86">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</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/>

<span class="kw">thus </span><a NAME="E2:157_1_1_1"/>
 <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:157_1_1"><span class="lab"><font color="Green" title="E88">A4</font></span></a>, <a class="txt" href="#E7:157_1_1"><span class="lab"><font color="Green" title="E89">A5</font></span></a>, <a class="txt" href="#E3:157_1_1"><span class="lab"><font color="Green" title="E86">A2</font></span></a>, <a class="ref" href="xxreal_3.html#D3" target="_self" title="XXREAL_3:def.3">Def3</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><span class="kw">hence </span><a NAME="E9:157_1_1"/>
<span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:157_1_1"><span class="lab"><font color="Green" title="E86">A2</font></span></a>, <a class="ref" href="xxreal_3.html#D6" target="_self" title="XXREAL_3:def.6">Def6</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="157_1_2"><span class="kw">suppose </span></a><a NAME="E1:157_1_2"/><span class="lab"><font color="Green" title="E85">A6</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</a></span>)</span></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</a>  = 
<a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#D3" target="_self" title="XXREAL_3:def.3">Def3</a></span>
<br/>.= 
 <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:157_1_2"><span class="lab"><font color="Green" title="E85">A6</font></span></a></span>
;<br/> <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="157_1_3"><span class="kw">suppose </span></a><a NAME="E1:157_1_3"/><span class="lab"><font color="Green" title="E85">A7</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</a></span>)</span></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</a>  = 
<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#D3" target="_self" title="XXREAL_3:def.3">Def3</a></span>
<br/>.= 
 <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K5" title="XXREAL_3:func.5">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:157_1_3"><span class="lab"><font color="Green" title="E85">A7</font></span></a></span>
;<br/> <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>
