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<b>let </b><font color="Maroon" title="c2">S</font> be   <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font>, <font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c6">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>;<br/>



<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font>
 ;<br/>

<b>assume </b><a NAME="E2:33"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">s</font>
 ;<br/>

<a NAME="E3:33"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">s</font> )
 
<b>by </b><i><a class="txt" href="sub_metr.html#E1:33"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="sub_metr.html#D7" target="_self" title="SUB_METR:def.7">Def7</a></i>;<br/>
<a NAME="E4:33"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sub_metr.html#E1:33"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="sub_metr.html#D7" target="_self" title="SUB_METR:def.7">Def7</a></i>;<br/>
<a NAME="E5:33"/><b>then </b><i><font color="Green" title="E22">A4</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">s</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sub_metr.html#E2:33"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="sub_metr.html#D7" target="_self" title="SUB_METR:def.7">Def7</a></i>;<br/>
<a NAME="E6:33"/><i><font color="Green" title="E23">A5</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/>
<a NAME="E7:33"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">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" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">s</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:33"/><b>then </b><i><font color="Green" title="E24">A6</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">s</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sub_metr.html#E5:33"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="sub_metr.html#E6:33"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E10:33"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">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"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font>
 ;<br/>

<b>hence </b><a NAME="E2:33_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sub_metr.html#E3:33"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="sub_metr.html#E5:33"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="metric_1.html#T2" title="METRIC_1:th.2">METRIC_1:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:33"/>
( <font color="Maroon" title="c4">q</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c5">r</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">s</font> )
 <b>by </b><i><a class="txt" href="sub_metr.html#E3:33"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="sub_metr.html#E5:33"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="sub_metr.html#E9:33"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="sub_metr.html#D7" target="_self" title="SUB_METR:def.7">Def7</a></i>;<br/>


</div>
