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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:21"/><i><font color="Green">E11</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> implies  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:21_1"/><i><font color="Green">E12</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:21_1"/><i><font color="Green">E13</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:21_1"/><i><font color="Green">E14</font></i>: 
0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="absvalue.html#E1:21_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="absvalue.html#E2:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E4:21_1"/><i><font color="Green">E15</font></i>: 
 <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="absvalue.html#E1:21_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E5:21_1"/>
 <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="absvalue.html#E2:21_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E6:21_1"/>
 <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E3:21_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="absvalue.html#E4:21_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:21"/><i><font color="Green">E12</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 implies  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:21_2"/><i><font color="Green">E13</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:21_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<a NAME="E3:21_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="absvalue.html#E1:21_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="absvalue.html#E2:21_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<a NAME="E4:21_2"/>
 <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i><a class="txt" href="absvalue.html#E2:21_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self">Def2</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="absvalue.html#E1:21_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self">Def2</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> 1

;<br/>
<b>hence </b><a NAME="E6:21_2"/>
 <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E3:21_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:21"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> implies  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:21_3"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>and </b><br/><a NAME="E2:21_3"/><i><font color="Green">E15</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:21_3"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="absvalue.html#E1:21_3"><i><font color="Green">E14</font></i></a>, <a class="txt" href="absvalue.html#E2:21_3"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E4:21_3"/>
 <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="absvalue.html#E2:21_3"><i><font color="Green">E15</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E5:21_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i><a class="txt" href="absvalue.html#E1:21_3"><i><font color="Green">E14</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E6:21_3"/>
 <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E3:21_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:21"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 implies  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:21_4"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>and </b><br/><a NAME="E2:21_4"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<a NAME="E3:21_4"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="absvalue.html#E1:21_4"><i><font color="Green">E15</font></i></a>, <a class="txt" href="absvalue.html#E2:21_4"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<a NAME="E4:21_4"/>
 <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i><a class="txt" href="absvalue.html#E2:21_4"><i><font color="Green">E16</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self">Def2</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="absvalue.html#E1:21_4"><i><font color="Green">E15</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self">Def2</a></i>
<br/>.= 
1

;<br/>
<b>hence </b><a NAME="E6:21_4"/>
 <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E3:21_4"><i><font color="Green">E17</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:21"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) implies  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_5"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:21_5"/><b>then </b>
(  <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or  <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="absvalue.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E3:21_5"/>
 <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E1:21_5"><i><font color="Green">E15</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:21"/>
 <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E1:21"><i><font color="Green">E11</font></i></a>, <a class="txt" href="absvalue.html#E2:21"><i><font color="Green">E12</font></i></a>, <a class="txt" href="absvalue.html#E3:21"><i><font color="Green">E13</font></i></a>, <a class="txt" href="absvalue.html#E4:21"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div>
