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<div class="add">

<b>reconsider </b><font color="Maroon">c<sub>1</sub></font> =  <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <a href="numbers.html#K1">REAL</a> ,<a href="metric_1.html#K7" target="_self">real_dist</a>  #) as  non <a href="struct_0.html#V3">empty</a>  <a href="metric_1.html#L1" target="_self">MetrStruct</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E2:31_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is   <a href="metric_1.html#NM1">MetrSpace</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_1_1_1"/>
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 <br/>( (  <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> 0 implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> implies  <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> 0 ) &amp;  <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> &amp;  <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> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T9" target="_self">Th9</a>, <a class="ref" href="metric_1.html#T10" target="_self">Th10</a>, <a class="ref" href="metric_1.html#T11" target="_self">Th11</a></i>;<br/>
<b>hence </b><a NAME="E2:31_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is   <a href="metric_1.html#NM1">MetrSpace</a>
 <b>by </b><i><a class="ref" href="metric_1.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:31_1_1"/>
(  <a href="metric_1.html#K8" target="_self">RealSpace</a>  is <a href="metric_1.html#V6" target="_self">Reflexive</a> &amp;  <a href="metric_1.html#K8" target="_self">RealSpace</a>  is <a href="metric_1.html#V7" target="_self">discerning</a> &amp;  <a href="metric_1.html#K8" target="_self">RealSpace</a>  is <a href="metric_1.html#V8" target="_self">symmetric</a> &amp;  <a href="metric_1.html#K8" target="_self">RealSpace</a>  is <a href="metric_1.html#V9" target="_self">triangle</a> )
 ;<br/>


</div>
