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<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">R</font> being  <a href="alg_1.html#NM2" title="ALG_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">R</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K3" title="TAXONOM1:func.3">dist_toler</a> <font color="Maroon" title="c1">M</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  holds <br/> <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K10" title="EQREL_1:func.10">SmallestPartition</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span><br/><b>let </b><font color="Maroon" title="c2">R</font> be   <a href="alg_1.html#NM2" title="ALG_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">R</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K3" title="TAXONOM1:func.3">dist_toler</a> <font color="Maroon" title="c1">M</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  implies  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K10" title="EQREL_1:func.10">SmallestPartition</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> )</span><br/>



<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K3" title="TAXONOM1:func.3">dist_toler</a> <font color="Maroon" title="c1">M</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K10" title="EQREL_1:func.10">SmallestPartition</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><a NAME="E1:46_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="metric_1.html#T5" title="METRIC_1:th.5">METRIC_1:5</a></i>;<br/><b>hence </b><a NAME="E2:46_1"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E3:46"/><b>then </b><i><font color="Green" title="E39">A2</font></i>: 
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> is <a href="taxonom1.html#V1" title="TAXONOM1:attr.1">nonnegative</a>
 
<b>by </b><i><a class="ref" href="taxonom1.html#D4" target="_self" title="TAXONOM1:def.4">Def4</a></i>;<br/>
<a NAME="E4:46"/><i><font color="Green" title="E40">A3</font></i>: 
( the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a> &amp; the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V3" title="METRIC_1:attr.3">discerning</a> )
 
<b>by </b><i><a class="ref" href="metric_1.html#D7" title="METRIC_1:def.7">METRIC_1:def 7</a>, <a class="ref" href="metric_1.html#D8" title="METRIC_1:def.8">METRIC_1:def 8</a></i>;<br/>
<a NAME="E5:46"/>
<span class="p1">(<span class="default"><a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="txt" href="#E1:46"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="taxonom1.html#T33" target="_self" title="TAXONOM1:th.33">Th33</a></i>;<br/>
<b>hence </b><a NAME="E6:46"/>
 <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K10" title="EQREL_1:func.10">SmallestPartition</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="txt" href="#E3:46"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="#E4:46"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="taxonom1.html#T24" target="_self" title="TAXONOM1:th.24">Th24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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