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<a NAME="E1:55_1_1"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# 1,<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a> #) holds   <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T23" target="_self" title="METRIC_1:th.23">Th24</a></span>;<br/>
<a NAME="E2:55_1_1"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# 1,<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a> #)  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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">x</font>,<font color="Olive" title="b2">y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T21" target="_self" title="METRIC_1:th.21">Th22</a></span>;<br/>
<a NAME="E3:55_1_1"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# 1,<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a> #) holds   <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="metric_1.html#E3"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/>
<a NAME="E4:55_1_1"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# 1,<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a> #) holds   <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T22" target="_self" title="METRIC_1:th.22">Th23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:55_1_1"/>
(  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# 1,<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a> #) is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>  &amp;  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# 1,<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a> #) is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  &amp;  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# 1,<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a> #) is  <a href="metric_1.html#V11" title="METRIC_1:attr.11">Discerning</a>  &amp;  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# 1,<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a> #) is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="#E1:55_1_1"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E3:55_1_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="metric_1.html#T25" target="_self" title="METRIC_1:th.25">Th26</a>, <a class="ref" href="metric_1.html#T1" target="_self" title="METRIC_1:th.1">Th1</a>, <a class="ref" href="metric_1.html#T3" target="_self" title="METRIC_1:th.3">Th3</a>, <a class="ref" href="metric_1.html#T4" target="_self" title="METRIC_1:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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