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<div class="add">

<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="E1:16_1"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>
 ;<br/><a NAME="E2:16_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#V4" target="_self">symmetric</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_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> 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#D5" target="_self">METRIC_1:def 5</a><br/>

<b>thus </b>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="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 href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="metric_1.html#E1:16_1"><i><font color="Green">E16</font></i></a></i>
<br/>.= 
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>2</sub></font>

;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="metric_1.html#V8" target="_self">symmetric</a>
 <b>by </b><i><a class="ref" href="metric_1.html#D9" target="_self">Def9</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:16"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="metric_1.html#V8" target="_self">symmetric</a>
 ;<br/>

<a NAME="E3:16"/><b>then </b><i><font color="Green">E16</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#V4" target="_self">symmetric</a>
 
<b>by </b><i><a class="ref" href="metric_1.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E4:16"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="metric_1.html#E3:16"><i><font color="Green">E16</font></i></a>, <a class="ref" href="metric_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E5:16"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>
 ;<br/>


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