<?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/>



<b>assume </b><b>that </b><br/><a NAME="E1:12"/><i><font color="Green">E16</font></i>: 
 <a href="complex1.html#K18">abs</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> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">y</font></span>)</span>
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green">E17</font></i>: 
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E3:12"/><i><font color="Green">E18</font></i>: 
(  not <font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</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="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> ) or ( 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 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<a NAME="E2:12_1"/><b>then </b><i><font color="Green">E21</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
;<br/>
<a NAME="E3:12_1"/><i><font color="Green">E22</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 0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<a NAME="E4:12_1"/>
( 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 <font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<b>hence </b><a NAME="E5:12_1"/>
( ( <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> ) or ( 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 ) )
 <b>by </b><i>, , <a class="ref" href="absvalue.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:12"/><i><font color="Green">E26</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> &amp; <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 implies  <a href="complex1.html#K18">abs</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#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:12_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>and </b><br/><a NAME="E2:12_2"/><i><font color="Green">E28</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E3:12_2"/><i><font color="Green">E29</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
 ;<br/>

<a NAME="E4:12_2"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" target="_self">Def1</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E5:12_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" target="_self">Def1</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E6:12_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:12_2"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3">*</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="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font>
 
;<br/>
<a NAME="E8:12_2"/><i><font color="Green">E31</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:12_2"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>hence </b><a NAME="E10:12_2"/>
 <a href="complex1.html#K18">abs</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#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">y</font></span>)</span>
 <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="txt" href="absvalue.html#E1:12"><i><font color="Green">E16</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:12"/><i><font color="Green">E32</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> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> implies  <a href="complex1.html#K18">abs</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#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:12_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>and </b><br/><a NAME="E2:12_3"/><i><font color="Green">E34</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E3:12_3"/><i><font color="Green">E35</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E4:12_3"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" target="_self">Def1</a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/>
<a NAME="E5:12_3"/><b>then </b>
<font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" target="_self">Def1</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E6:12_3"/><b>then </b><i><font color="Green">E36</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> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:12_3"/><i><font color="Green">E37</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E8:12_3"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="absvalue.html#E3"><i><font color="Green">Lemma14</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E9:12_3"/>
 <a href="complex1.html#K18">abs</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#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E4"><i><font color="Green">Lemma15</font></i></a>, <a class="txt" href="absvalue.html#E1:12"><i><font color="Green">E16</font></i></a>, <a class="txt" href="absvalue.html#E2:12"><i><font color="Green">E17</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:12"/><i><font color="Green">E38</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 &amp; <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 implies  <a href="complex1.html#K18">abs</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#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:12_4"/><i><font color="Green">E39</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E2:12_4"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>and </b><br/><a NAME="E3:12_4"/><i><font color="Green">E41</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
 ;<br/>

<a NAME="E4:12_4"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 
<b>by </b><i><a class="txt" href="absvalue.html#E3"><i><font color="Green">Lemma14</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E5:12_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="absvalue.html#E3"><i><font color="Green">Lemma14</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E6:12_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:12_4"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3">*</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="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">y</font></span>)</span>
 
;<br/>
<a NAME="E8:12_4"/><i><font color="Green">E43</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="absvalue.html#E3"><i><font color="Green">Lemma14</font></i></a>, </i>;<br/>
<a NAME="E9:12_4"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="absvalue.html#E4"><i><font color="Green">Lemma15</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E10:12_4"/>
 <a href="complex1.html#K18">abs</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#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E1:12"><i><font color="Green">E16</font></i></a>, <a class="txt" href="absvalue.html#E2:12"><i><font color="Green">E17</font></i></a>, <a class="txt" href="absvalue.html#E3:12"><i><font color="Green">E18</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:12"/>
( 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 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> implies  <a href="complex1.html#K18">abs</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#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:12_5"/><i><font color="Green">E44</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E2:12_5"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>and </b><br/><a NAME="E3:12_5"/><i><font color="Green">E46</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E4:12_5"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="absvalue.html#E4"><i><font color="Green">Lemma15</font></i></a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/>
<a NAME="E5:12_5"/><b>then </b>
0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
 
<b>by </b><i><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="E6:12_5"/><b>then </b><i><font color="Green">E47</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> <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:12_5"/><i><font color="Green">E48</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="absvalue.html#E3"><i><font color="Green">Lemma14</font></i></a>, </i>;<br/>
<a NAME="E8:12_5"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="absvalue.html#E4"><i><font color="Green">Lemma15</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E9:12_5"/>
 <a href="complex1.html#K18">abs</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#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="absvalue.html#E1:12"><i><font color="Green">E16</font></i></a>, <a class="txt" href="absvalue.html#E2:12"><i><font color="Green">E17</font></i></a>, <a class="txt" href="absvalue.html#E3:12"><i><font color="Green">E18</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:12"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="absvalue.html#D1" target="_self">Def1</a></i>;<br/>


</div>
