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

<span class="kw">let </span><font color="Maroon" title="c1">x</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">x</font></span>)</span></span><br/>

<a NAME="E1:40"/><span class="lab"><font color="Green" title="E7">A1</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> <font color="Maroon" title="c1">x</font> implies  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">x</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:40_1"/><span class="lab"><font color="Green" title="E8">A2</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> <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">  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">x</font></span>)</span></span><br/>

<a NAME="E2:40_1"/>
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T39" target="_self" title="ABSVALUE:th.39">Th39</a></span>;<br/>
<span class="kw">then </span> <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">x</font></span>)</span> = 
1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></span>
<br/>.= 
1

;<br/>
<span class="kw">hence </span><a NAME="E4:40_1"/>
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</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>
<a NAME="E2:40"/>
( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">x</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:40_2"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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">  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">x</font></span>)</span></span><br/>

<a NAME="E2:40_2"/><span class="kw">then </span>
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></span>;<br/>
<a NAME="E3:40_2"/><span class="kw">then </span>
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T39" target="_self" title="ABSVALUE:th.39">Th39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:40_2"/>
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_2"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</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>
<span class="kw">hence </span><a NAME="E3:40"/>
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E7">A1</font></span></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>
