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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/><b>let </b><font color="Maroon">p1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Maroon">p2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Maroon">q1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Maroon">q2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 ;<br/>

<a NAME="E2:28"/>
<font color="Maroon">V</font> is    <a href="metric_2.html#M1" target="_self">equivalence_class</a> of <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:28"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:28"/><b>then </b><i><font color="Green">E29</font></i>: 
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:28"/>
<font color="Maroon">Q</font> is    <a href="metric_2.html#M1" target="_self">equivalence_class</a> of <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:28"/><b>then </b>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E7:28"/><b>then </b><i><font color="Green">E37</font></i>: 
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:28"/><i><font color="Green">E52</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</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">p2</font>,<font color="Maroon">q1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>
<a NAME="E9:28"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p2</font>,<font color="Maroon">q1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">p2</font>,<font color="Maroon">q2</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">q2</font>,<font color="Maroon">q1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>
<a NAME="E10:28"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p2</font>,<font color="Maroon">q2</font>
 
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:28"/><i><font color="Green">E54</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p2</font>,<font color="Maroon">q2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</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">p1</font>,<font color="Maroon">q2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>
<a NAME="E12:28"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p1</font>,<font color="Maroon">q2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</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">q1</font>,<font color="Maroon">q2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>
<a NAME="E13:28"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p2</font>,<font color="Maroon">q2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</font>
 
<b>by </b><i>, , <a class="ref" href="metric_2.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E14:28"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p2</font>,<font color="Maroon">q2</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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