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<b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="metric_1.html#L1">MetrStruct</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E13</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>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/>0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="metric_1.html#K2">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>
 ;<br/><a NAME="E2:12_1"/>
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="sub_metr.html#V1" target="_self">Discerning</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>proof </b></a><div class="add">

<b>let </b><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>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="sub_metr.html#D1" target="_self">SUB_METR:def 1</a><br/>

<b>assume </b><a NAME="E1:12_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:12_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="sub_metr.html#E1:12_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:12_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:12_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="sub_metr.html#V2" target="_self">Discerning</a>
 <b>by </b><i><a class="ref" href="sub_metr.html#D2" target="_self">Def2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:12"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="sub_metr.html#V2" target="_self">Discerning</a>
 ;<br/>

<a NAME="E3:12"/><b>then </b><i><font color="Green">E13</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="sub_metr.html#V1" target="_self">Discerning</a>
 
<b>by </b><i><a class="ref" href="sub_metr.html#D2" target="_self">Def2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><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> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:12_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:12_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="sub_metr.html#E3:12"><i><font color="Green">E13</font></i></a>, <a class="ref" href="sub_metr.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:12_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:12"/>
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>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/>0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="metric_1.html#K2">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>
 ;<br/>


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