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

<b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">y</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:26"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:26_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>and </b><br/><a NAME="E2:26_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<a NAME="E3:26_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i><a class="txt" href="absvalue.html#E3"><i><font color="Green">Lemma14</font></i></a>, <a class="txt" href="absvalue.html#E4"><i><font color="Green">Lemma15</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E4:26_1"/><i><font color="Green">E21</font></i>: 
 <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</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#E3"><i><font color="Green">Lemma14</font></i></a>, </i>;<br/>
<a NAME="E5:26_1"/><i><font color="Green">E22</font></i>: 
 <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</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#E4"><i><font color="Green">Lemma15</font></i></a>, </i>;<br/>
<a NAME="E6:26_1"/><i><font color="Green">E26</font></i>: 
 <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="absvalue.html#E1:26"><i><font color="Green">E16</font></i></a>, <a class="txt" href="absvalue.html#E1:26_1"><i><font color="Green">E17</font></i></a>, </i>;<br/>
<a NAME="E7:26_1"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&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" target="_self">sgn</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E8:26_1"/>
<span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E9:26_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E2:26_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="absvalue.html#E13"><i><font color="Green">Lemma19</font></i></a>, <a class="txt" href="absvalue.html#E3:26_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:26"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">x</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">y</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:26_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>and </b><br/><a NAME="E2:26_2"/><i><font color="Green">E30</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E3:26_2"/>
 <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</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#E4"><i><font color="Green">Lemma15</font></i></a>, </i>;<br/>
<b>then </b><i><font color="Green">E31</font></i>: <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="absvalue.html#E1:26"><i><font color="Green">E16</font></i></a>, </i>
<br/>.= 
0

;<br/>
<a NAME="E5:26_2"/>
( <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E6:26_2"/><b>then </b>
(  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 or  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:26_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:26"/><i><font color="Green">E32</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:26_3"/><i><font color="Green">E33</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E2:26_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<a NAME="E3:26_3"/>
 <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="absvalue.html#E1:26"><i><font color="Green">E16</font></i></a>, </i>;<br/>
<b>then </b><i><font color="Green">E35</font></i>: <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span> = 
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
0

;<br/>
<a NAME="E5:26_3"/>
( <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E6:26_3"/><b>then </b>
(  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 or  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:26_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:26"/><i><font color="Green">E36</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:26_4"/><i><font color="Green">E37</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E2:26_4"/><i><font color="Green">E38</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E3:26_4"/><i><font color="Green">E39</font></i>: 
0 <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E4:26_4"/>
 <a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:26_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:26_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E13"><i><font color="Green">Lemma19</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:26"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 ) implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_5"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:26_5"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_5_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>then </b><i><font color="Green">E43</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 = 
<span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1

;<br/>
<a NAME="E3:26_5_1"/>
<span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E4:26_5_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E13"><i><font color="Green">Lemma19</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:26_5"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_5_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>then </b><i><font color="Green">E45</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1

;<br/>
<a NAME="E3:26_5_2"/>
<span class="p1">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E4:26_5_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E13"><i><font color="Green">Lemma19</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:26_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:26"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="absvalue.html#K1" target="_self">sgn</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="absvalue.html#K1" target="_self">sgn</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1" target="_self">Def1</a>, <a class="txt" href="absvalue.html#E3"><i><font color="Green">Lemma14</font></i></a>, <a class="txt" href="absvalue.html#E4"><i><font color="Green">Lemma15</font></i></a>, <a class="txt" href="absvalue.html#E1:26"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div>
