<?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> ;<br/>



<a NAME="E1:25"/><i><font color="Green" title="E8">A1</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 <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:25_1"/><i><font color="Green" title="E9">A2</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="E2:25_1"/><i><font color="Green" title="E10">A3</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> 
 ;<br/>

<a NAME="E3:25_1"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="absvalue.html#E1:25_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="absvalue.html#E2:25_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E4:25_1"/><i><font color="Green" title="E12">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>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<b>by </b><i><a class="txt" href="absvalue.html#E1:25_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>;<br/>
<a NAME="E5:25_1"/><i><font color="Green" title="E13">A6</font></i>: 
 <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="absvalue.html#E2:25_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>;<br/>
<a NAME="E6:25_1"/><i><font color="Green" title="E14">A7</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>  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="absvalue.html#E3:25_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="absvalue.html#E4:25_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>;<br/>
<a NAME="E7:25_1"/><i><font color="Green" title="E15">A8</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E8:25_1"/>
<span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E9:25_1"/>
<span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E5:25_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="txt" href="absvalue.html#E6:25_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="absvalue.html#E7:25_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:25"/><i><font color="Green" title="E9">A9</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 <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:25_2"/><i><font color="Green" title="E10">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> 
 <b>and </b><br/><a NAME="E2:25_2"/><i><font color="Green" title="E11">A11</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>
 ;<br/>

<a NAME="E3:25_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="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<b>by </b><i><a class="txt" href="absvalue.html#E1:25_2"><i><font color="Green" title="E10">A10</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>;<br/>
<b>then </b><i><font color="Green" title="E12">A12</font></i>: <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#K2" title="XCMPLX_0:func.2">+</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#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="absvalue.html#E2:25_2"><i><font color="Green" title="E11">A11</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/>
<a NAME="E5:25_2"/>
( <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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>  or <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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>  or  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font> )
 
;<br/>
<b>hence </b><a NAME="E6:25_2"/>
<span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E4:25_2"><i><font color="Green" title="E12">A12</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:25"/><i><font color="Green" title="E10">A13</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 <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:25_3"/><i><font color="Green" title="E11">A14</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="E2:25_3"/><i><font color="Green" title="E12">A15</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> 
 ;<br/>

<a NAME="E3:25_3"/>
 <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="absvalue.html#E1:25_3"><i><font color="Green" title="E11">A14</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>;<br/>
<b>then </b><i><font color="Green" title="E13">A16</font></i>: <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#K2" title="XCMPLX_0:func.2">+</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#K7" title="REAL_1:func.7">+</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="absvalue.html#E2:25_3"><i><font color="Green" title="E12">A15</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/>
<a NAME="E5:25_3"/>
( <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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>  or <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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>  or  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font> )
 
;<br/>
<b>hence </b><a NAME="E6:25_3"/>
<span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E4:25_3"><i><font color="Green" title="E13">A16</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:25"/><i><font color="Green" title="E11">A17</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 <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:25_4"/><i><font color="Green" title="E12">A18</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="E2:25_4"/><i><font color="Green" title="E13">A19</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>
 ;<br/>

<a NAME="E3:25_4"/><i><font color="Green" title="E14">A20</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="txt" href="absvalue.html#E1:25_4"><i><font color="Green" title="E12">A18</font></i></a>, <a class="txt" href="absvalue.html#E2:25_4"><i><font color="Green" title="E13">A19</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E4:25_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> 1
 
<b>by </b><i><a class="txt" href="absvalue.html#E2:25_4"><i><font color="Green" title="E13">A19</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>;<br/>
<a NAME="E5:25_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="absvalue.html#E1:25_4"><i><font color="Green" title="E12">A18</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="E6:25_4"/>
<span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E3:25_4"><i><font color="Green" title="E14">A20</font></i></a>, <a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:25"/>
( ( <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 <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_5"/><i><font color="Green" title="E12">A21</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>  )
 ;<br/>

<a NAME="E2:25_5"/><i><font color="Green" title="E13">A22</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>  implies <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_5_1"/><i><font color="Green" title="E14">A23</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> 
 ;<br/>

<b>then </b><i><font color="Green" title="E15">A24</font></i>: <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 = 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<b>by </b><i><a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>
<br/>.= 
<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="real_1.html#K5" title="REAL_1:func.5">-</a> 1

;<br/>
<a NAME="E3:25_5_1"/>
<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="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<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="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E4:25_5_1"/>
<span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E1:25_5_1"><i><font color="Green" title="E14">A23</font></i></a>, <a class="txt" href="absvalue.html#E2:25_5_1"><i><font color="Green" title="E15">A24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:25_5"/>
( <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 <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_5_2"/><i><font color="Green" title="E14">A25</font></i>: 
<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> 
 ;<br/>

<b>then </b><i><font color="Green" title="E15">A26</font></i>: <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<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="real_1.html#K3" title="REAL_1:func.3">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<b>by </b><i><a class="ref" href="absvalue.html#D2" target="_self" title="ABSVALUE:def.2">Def2</a></i>
<br/>.= 
<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="real_1.html#K5" title="REAL_1:func.5">-</a> 1

;<br/>
<a NAME="E3:25_5_2"/>
<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="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<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="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E4:25_5_2"/>
<span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E1:25_5_2"><i><font color="Green" title="E14">A25</font></i></a>, <a class="txt" href="absvalue.html#E2:25_5_2"><i><font color="Green" title="E15">A26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:25_5"/>
<span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E1:25_5"><i><font color="Green" title="E12">A21</font></i></a>, <a class="txt" href="absvalue.html#E2:25_5"><i><font color="Green" title="E13">A22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:25"/>
<span class="p1">(<span class="default"><span class="p2">(<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#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E1:25"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="absvalue.html#E2:25"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="absvalue.html#E3:25"><i><font color="Green" title="E10">A13</font></i></a>, <a class="txt" href="absvalue.html#E4:25"><i><font color="Green" title="E11">A17</font></i></a></i>;<br/>


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