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

<b>let </b><font color="Maroon">S</font> be   <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>



<b>assume </b><a NAME="E1:33"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">q</font> <a href="sub_metr.html#R1" target="_self">is_between</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 ;<br/>

<b>assume </b><a NAME="E2:33"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">r</font> <a href="sub_metr.html#R1" target="_self">is_between</a> <font color="Maroon">p</font>,<font color="Maroon">s</font>
 ;<br/>

<a NAME="E3:33"/><b>then </b><i><font color="Green">E26</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:33"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:33"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:33"/><i><font color="Green">E56</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">q</font>,<font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>
<a NAME="E7:33"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">q</font>,<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>
<a NAME="E8:33"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">q</font>,<font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:33"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">q</font>,<font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E10:33"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">q</font>,<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 
;<br/>
<a NAME="E11:33"/>
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E2:33_1"/><i><font color="Green">E65</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">q</font>,<font color="Maroon">r</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>
<a NAME="E3:33_1"/>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="sub_metr.html#T1" target="_self">Th1</a>, <a class="ref" href="metric_1.html#T2">METRIC_1:2</a></i>;<br/>
<a NAME="E4:33_1"/><b>then </b>
0 <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sub_metr.html#E3:33"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E5:33_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sub_metr.html#E1:33"><i><font color="Green">E24</font></i></a>, <a class="txt" href="sub_metr.html#E2:33"><i><font color="Green">E25</font></i></a>, <a class="ref" href="metric_1.html#T1">METRIC_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:33"/>
( <font color="Maroon">q</font> <a href="sub_metr.html#R1" target="_self">is_between</a> <font color="Maroon">p</font>,<font color="Maroon">s</font> &amp; <font color="Maroon">r</font> <a href="sub_metr.html#R1" target="_self">is_between</a> <font color="Maroon">q</font>,<font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="sub_metr.html#T1" target="_self">Th1</a>, , , <a class="txt" href="sub_metr.html#E1:33"><i><font color="Green">E24</font></i></a>, </i>;<br/>


</div>
