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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="metric_1.html#L1" target="_self">MetrStruct</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span>
 ;<br/><a NAME="E2:17_1"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="metric_1.html#V5" target="_self">triangle</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D6" target="_self">METRIC_1:def 6</a><br/>

<a NAME="E1:17_1_1"/><i><font color="Green">E18</font></i>: 
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E2:17_1_1"/><i><font color="Green">E19</font></i>: 
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<a NAME="E3:17_1_1"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1" target="_self">.</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" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<b>hence </b><a NAME="E4:17_1_1"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="metric_1.html#E1:17_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="metric_1.html#E1:17_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="metric_1.html#E2:17_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:17_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="metric_1.html#V9" target="_self">triangle</a>
 <b>by </b><i><a class="ref" href="metric_1.html#D10" target="_self">Def10</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:17"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="metric_1.html#V9" target="_self">triangle</a>
 ;<br/>

<a NAME="E3:17"/><b>then </b><i><font color="Green">E17</font></i>: 
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="metric_1.html#V5" target="_self">triangle</a>
 
<b>by </b><i><a class="ref" href="metric_1.html#D10" target="_self">Def10</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>thus </b><a NAME="E4:17"/>
 <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="metric_1.html#E3:17"><i><font color="Green">E17</font></i></a>, <a class="ref" href="metric_1.html#D6" target="_self">Def6</a></i>;<br/>


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