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<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>;<br/>



<b>assume </b><a NAME="E1:32"/><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/>

<a NAME="E2:32"/><b>then </b><i><font color="Green">E25</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">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:32"/><i><font color="Green">E26</font></i>: 
 <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/>
<b>thus </b><a NAME="E4:32"/>
not <font color="Maroon">p</font> <a href="sub_metr.html#R1" target="_self">is_between</a> <font color="Maroon">q</font>,<font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">p</font> <a href="sub_metr.html#R1" target="_self">is_between</a> <font color="Maroon">q</font>,<font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:32_1"/><b>then </b>
 <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"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">q</font>,<font color="Maroon">p</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">p</font>,<font color="Maroon">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sub_metr.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E3:32_1"/><b>then </b>
 <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"><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">p</font>,<font color="Maroon">q</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">r</font></span>)</span>
 
;<br/>
<a NAME="E4:32_1"/><b>then </b>
0 <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">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <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></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">r</font></span>)</span>
 
;<br/>
<a NAME="E5:32_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</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>
 
;<br/>
<a NAME="E6:32_1"/><b>then </b>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="hidden.html#R1">=</a> 2 )
 
;<br/>
<b>hence </b><a NAME="E7:32_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="metric_1.html#T2">METRIC_1:2</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E5:32"/>
<font color="Maroon">r</font> <a href="sub_metr.html#R1" target="_self">is_between</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 ;<br/>

<a NAME="E6:32"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</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">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sub_metr.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E7:32"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</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"><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">q</font>,<font color="Maroon">r</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E8:32"/><b>then </b>
0 <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> <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">2 <a href="xcmplx_0.html#K3">*</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>
 
;<br/>
<a NAME="E9:32"/><b>then </b>
0 <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</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>
 
;<br/>
<a NAME="E10:32"/><b>then </b>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="hidden.html#R1">=</a> 2 )
 
;<br/>
<b>hence </b><a NAME="E11:32"/>
contradiction
 <b>by </b><i>, <a class="ref" href="metric_1.html#T2">METRIC_1:2</a></i>;<br/>


</div>
