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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> ;<br/>



<a NAME="E1:51"/><i><font color="Green">E28</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> 0 implies <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1"/><i><font color="Green">E29</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> 0
 ;<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="E3:51_1"/><i><font color="Green">E37</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="E4:51_1"/><i><font color="Green">E52</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#E1:51"><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="E6:51_1"/><i><font color="Green">E53</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="E7:51_1"/>
<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/>
<a NAME="E8:51_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
 
<b>by </b><i><a class="ref" href="metric_2.html#T6" target="_self">Th6</a>, <a class="txt" href="metric_2.html#E1:51_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="metric_2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E9:51_1"/><b>then </b>
<font color="Maroon">p</font> <a href="metric_2.html#R3" target="_self">tolerates</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:51_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="txt" href="metric_2.html#E1:51"><i><font color="Green">E28</font></i></a>, <a class="ref" href="metric_2.html#T7" target="_self">Th7</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:51"/>
( <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> implies <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> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 ;<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="E3:51_2"/><i><font color="Green">E74</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="E4:51_2"/><i><font color="Green">E75</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#E1:51"><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="E6:51_2"/><i><font color="Green">E76</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="E7:51_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">p</font> <a href="metric_2.html#R3" target="_self">tolerates</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="metric_2.html#T6" target="_self">Th6</a>, , <a class="ref" href="metric_2.html#T7" target="_self">Th7</a>, </i>;<br/>
<a NAME="E8:51_2"/>
<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>then </b><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="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
<b>by </b><i>, <a class="ref" href="metric_2.html#D3" target="_self">Def3</a></i>
<br/>.= 
0
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E10:51_2"/>
<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> 0
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:51"/>
( <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> 0 iff <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="txt" href="metric_2.html#E4"><i><font color="Green">Lemma26</font></i></a></i>;<br/>


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