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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#V6">Reflexive</a> <a href="metric_1.html#V8">symmetric</a> <a href="metric_1.html#V9">triangle</a>  <a href="metric_1.html#L1">MetrStruct</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="E1:31"/><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="E2:31"/><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> &amp;  <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/>;<br/>
<b>thus </b><a NAME="E3:31"/>
( <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> )
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="sub_metr.html#D7" target="_self">SUB_METR:def 7</a><br/>

<b>thus </b><a NAME="E4:31"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">r</font>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">r</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#K2">dist</a> <font color="Maroon">q</font>,<font color="Maroon">p</font></span>)</span>
 <b>by </b><i/>;<br/>


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