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<b>let </b><font color="Maroon" title="c2">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:11_1"/><i><font color="Green" title="E13">A1</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> holds <br/>0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>
 ;<br/><a NAME="E2:11_1"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c2">M</font> is <a href="sub_metr.html#V1" title="SUB_METR:attr.1">Discerning</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="sub_metr.html#D1" target="_self" title="SUB_METR:def.1">SUB_METR:def 1</a><br/>

<b>assume </b><a NAME="E1:11_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 ;<br/>

<a NAME="E2:11_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 
<b>by </b><i><a class="txt" href="sub_metr.html#E1:11_1"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:11_1_1"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c2">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:11_1"/>
<font color="Maroon" title="c2">M</font> is <a href="sub_metr.html#V2" title="SUB_METR:attr.2">Discerning</a>
 <b>by </b><i><a class="ref" href="sub_metr.html#D2" target="_self" title="SUB_METR:def.2">Def2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:11"/>
<font color="Maroon" title="c2">M</font> is <a href="sub_metr.html#V2" title="SUB_METR:attr.2">Discerning</a>
 ;<br/>

<a NAME="E3:11"/><b>then </b><i><font color="Green" title="E13">A2</font></i>: 
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c2">M</font> is <a href="sub_metr.html#V1" title="SUB_METR:attr.1">Discerning</a>
 
<b>by </b><i><a class="ref" href="sub_metr.html#D2" target="_self" title="SUB_METR:def.2">Def2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font>;<br/><b>assume </b><a NAME="E1:11_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 ;<br/><a NAME="E2:11_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c2">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="sub_metr.html#E3:11"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="sub_metr.html#D1" target="_self" title="SUB_METR:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E3:11_2"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:11"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> holds <br/>0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>
 ;<br/>


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