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<b>let </b><font color="Maroon">M</font> be   <a href="sub_metr.html#NM1">PseudoMetricSpace</a>;<br/><b>let </b><font color="Maroon">V</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a> , <font color="Maroon">Q</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a> , <font color="Maroon">W</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a> ;<br/>



<b>consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E2:53"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 <b>by </b><i/>;<br/>
<a NAME="E3:53"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E2:53"><i><font color="Green">E28</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E5:53"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 <b>by </b><i/>;<br/>
<a NAME="E6:53"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="metric_2.html#T7" target="_self">Th7</a>, </i>;<br/>
<b>consider </b><font color="Maroon">w</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E8:53"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 <b>by </b><i/>;<br/>
<a NAME="E9:53"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:53"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="metric_2.html#K10" target="_self">nbourdist</a> <font color="Maroon">M</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">V</font>,<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E3:53"><i><font color="Green">E29</font></i></a>, , <a class="ref" href="metric_2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E11:53"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><a href="metric_2.html#K10" target="_self">nbourdist</a> <font color="Maroon">M</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">V</font>,<font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">w</font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E3:53"><i><font color="Green">E29</font></i></a>, <a class="ref" href="metric_2.html#T9" target="_self">Th9</a>, <a class="ref" href="metric_2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E12:53"/>
<span class="p1">(<span class="default"><a href="metric_2.html#K10" target="_self">nbourdist</a> <font color="Maroon">M</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">Q</font>,<font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">q</font>,<font color="Maroon">w</font>
 
<b>by </b><i>, <a class="ref" href="metric_2.html#T9" target="_self">Th9</a>, <a class="ref" href="metric_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E13:53"/>
<span class="p1">(<span class="default"><a href="metric_2.html#K10" target="_self">nbourdist</a> <font color="Maroon">M</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">V</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_2.html#K10" target="_self">nbourdist</a> <font color="Maroon">M</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">V</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_2.html#K10" target="_self">nbourdist</a> <font color="Maroon">M</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">Q</font>,<font color="Maroon">W</font></span>)</span>
 <b>by </b><i><a class="ref" href="metric_2.html#T12" target="_self">Th12</a>, <a class="ref" href="metric_2.html#T13" target="_self">Th13</a>, <a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>


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