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<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="metric_2.html#K2" target="_self">-neighbour</a> </span>)</span>,<span class="p1">(<span class="default"><a href="metric_2.html#K10" target="_self">nbourdist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="metric_2.html#K2" target="_self">-neighbour</a> </span>)</span>,<span class="p1">(<span class="default"><a href="metric_2.html#K10" target="_self">nbourdist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_2.html#K2" target="_self">-neighbour</a>  ;<br/><a NAME="E2:54_1_1_1"/><i><font color="Green">E28</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_2.html#K10" target="_self">nbourdist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E3:54_1_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_2.html#K10" target="_self">nbourdist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E4:54_1_1_1"/><i><font color="Green">E30</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_2.html#K10" target="_self">nbourdist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E5:54_1_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_2.html#K10" target="_self">nbourdist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><b>thus </b><a NAME="E6:54_1_1_1"/>
(  <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="metric_2.html#E2:54_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="metric_2.html#T54" target="_self">Th54</a></i>;<br/><b>thus </b><a NAME="E7:54_1_1_1"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="metric_2.html#E2:54_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="metric_2.html#E3:54_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="metric_2.html#T55" target="_self">Th55</a></i>;<br/><b>thus </b><a NAME="E8:54_1_1_1"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="metric_2.html#E2:54_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="metric_2.html#E4:54_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="metric_2.html#E5:54_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="metric_2.html#T56" target="_self">Th56</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:54_1_1"/>
 <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="metric_2.html#K2" target="_self">-neighbour</a> </span>)</span>,<span class="p1">(<span class="default"><a href="metric_2.html#K10" target="_self">nbourdist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is   <a href="metric_1.html#NM1">MetrSpace</a>
 <b>by </b><i><a class="ref" href="metric_1.html#T6">METRIC_1:6</a></i>;<br/>


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