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

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</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">  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/>



<a NAME="E1:18"/><i><font color="Green" title="E6">A1</font></i>: 
(  <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> &amp;  <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="c2">y</font> implies  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:18_1"/><i><font color="Green" title="E7">A2</font></i>: 
 <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>
 <b>and </b><br/><a NAME="E2:18_1"/><i><font color="Green" title="E8">A3</font></i>: 
 <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="c2">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/>

<a NAME="E3:18_1"/><i><font color="Green" title="E9">A4</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="txt" href="#E1:18_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="#E2:18_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E4:18_1"/><i><font color="Green" title="E10">A5</font></i>: 
 <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> 1
 
<b>by </b><i><a class="txt" href="#E1:18_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>;<br/>
<a NAME="E5:18_1"/>
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="#E2:18_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E6:18_1"/>
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:18_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="#E4:18_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</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="E2:18"/><i><font color="Green" title="E7">A6</font></i>: 
(  <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> &amp; <font color="Maroon" title="c2">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:18_2"/><i><font color="Green" title="E8">A7</font></i>: 
 <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>
 <b>and </b><br/><a NAME="E2:18_2"/><i><font color="Green" title="E9">A8</font></i>: 
<font color="Maroon" title="c2">y</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/>

<a NAME="E3:18_2"/><i><font color="Green" title="E10">A9</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</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 href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="txt" href="#E1:18_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="txt" href="#E2:18_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="xreal_1.html#T71" title="XREAL_1:th.71">XREAL_1:71</a></i>;<br/>
<a NAME="E4:18_2"/>
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<b>by </b><i><a class="txt" href="#E2:18_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>;<br/>
<b>then </b><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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span> = 
1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E1:18_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1

;<br/>
<b>hence </b><a NAME="E6:18_2"/>
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:18_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</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="E3:18"/><i><font color="Green" title="E8">A10</font></i>: 
( <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>  &amp;  <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="c2">y</font> implies  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:18_3"/><i><font color="Green" title="E9">A11</font></i>: 
<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> 
 <b>and </b><br/><a NAME="E2:18_3"/><i><font color="Green" title="E10">A12</font></i>: 
 <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="c2">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/>

<a NAME="E3:18_3"/><i><font color="Green" title="E11">A13</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</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 href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="txt" href="#E1:18_3"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="#E2:18_3"><i><font color="Green" title="E10">A12</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E4:18_3"/>
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="#E2:18_3"><i><font color="Green" title="E10">A12</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>;<br/>
<a NAME="E5:18_3"/><b>then </b>
<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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<b>by </b><i><a class="txt" href="#E1:18_3"><i><font color="Green" title="E9">A11</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E6:18_3"/>
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:18_3"><i><font color="Green" title="E11">A13</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</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="E4:18"/><i><font color="Green" title="E9">A14</font></i>: 
( <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>  &amp; <font color="Maroon" title="c2">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:18_4"/><i><font color="Green" title="E10">A15</font></i>: 
<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> 
 <b>and </b><br/><a NAME="E2:18_4"/><i><font color="Green" title="E11">A16</font></i>: 
<font color="Maroon" title="c2">y</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/>

<a NAME="E3:18_4"/><i><font color="Green" title="E12">A17</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <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 href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="txt" href="#E1:18_4"><i><font color="Green" title="E10">A15</font></i></a>, <a class="txt" href="#E2:18_4"><i><font color="Green" title="E11">A16</font></i></a>, <a class="ref" href="xreal_1.html#T71" title="XREAL_1:th.71">XREAL_1:71</a></i>;<br/>
<a NAME="E4:18_4"/>
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<b>by </b><i><a class="txt" href="#E2:18_4"><i><font color="Green" title="E11">A16</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>;<br/>
<b>then </b><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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E1:18_4"><i><font color="Green" title="E10">A15</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>
<br/>.= 
1

;<br/>
<b>hence </b><a NAME="E6:18_4"/>
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:18_4"><i><font color="Green" title="E12">A17</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</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="E5:18"/>
( ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_5"/><i><font color="Green" title="E10">A18</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/>

<a NAME="E2:18_5"/><b>then </b>
(  <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="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E3:18_5"/>
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:18_5"><i><font color="Green" title="E10">A18</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</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="E6:18"/>
 <a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="#E2:18"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="#E3:18"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="#E4:18"><i><font color="Green" title="E9">A14</font></i></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>
