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

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<a NAME="E1:27"/><i><font color="Green">E39</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<a NAME="E1:27_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">z</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E2:27_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T142">COMPLEX1:142</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:27"/>
( <a href="metric_1.html#K7" target="_self">real_dist</a>  <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y</font></span>)</span> &amp; <a href="metric_1.html#K7" target="_self">real_dist</a>  <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">z</font></span>)</span> &amp; <a href="metric_1.html#K7" target="_self">real_dist</a>  <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:27"/>
<a href="metric_1.html#K7" target="_self">real_dist</a>  <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K7" target="_self">real_dist</a>  <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K7" target="_self">real_dist</a>  <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">z</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="metric_1.html#E5"><i><font color="Green">Lemma23</font></i></a></i>;<br/>


</div>
