<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c2">M</font> =  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #);<br/>
<a NAME="E1:27_1_1"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) is  <a href="metric_1.html#V3" title="METRIC_1:attr.3">discerning</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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 <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #); <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="metric_1.html#D4" target="_self" title="METRIC_1:def.4">METRIC_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_1_1_1"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font></span><br/>

<span class="kw">hence </span><a NAME="E2:27_1_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:27_1_1"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) is  <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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 <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #); <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="metric_1.html#D5" target="_self" title="METRIC_1:def.5">METRIC_1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:27_1_1_2_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> or <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:27_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><br/><div class="add"><span class="kw">hence </span>the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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> = 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></span>
<br/>.= 
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:27_1_1_2_1_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:27_1_1_2_1_1_2"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:27_1_1_2"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:27_1_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) is  <a href="metric_1.html#V5" title="METRIC_1:attr.5">triangle</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be   <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>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #); <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="metric_1.html#D6" target="_self" title="METRIC_1:def.6">METRIC_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font></span>)</span></span><br/>

<a NAME="E1:27_1_1_3"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
( the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:27_1_1_3_1"/>
( ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> ) or ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font> ) or ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> &amp; <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> ) or ( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font> ) or ( <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> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:27_1_1_3_1_1"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:27_1_1_3_1_1"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_1_3"><span class="lab"><font color="Green" title="E21">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:27_1_1_3_1_2"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:27_1_1_3_1_2"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_1_3"><span class="lab"><font color="Green" title="E21">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_3_1_3"><span class="kw">suppose </span></a><a NAME="E1:27_1_1_3_1_3"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> &amp; <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font></span>)</span></span><br/><div class="add"><a NAME="E2:27_1_1_3_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A7</font></span>: 
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></span>;<br/><a NAME="E3:27_1_1_3_1_3"/>
( the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_1_3_1_3"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:27_1_1_3_1_3"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_1_3_1_3"><span class="lab"><font color="Green" title="E23">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_3_1_4"><span class="kw">suppose </span></a><a NAME="E1:27_1_1_3_1_4"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font></span>)</span></span><br/><div class="add"><a NAME="E2:27_1_1_3_1_4"/><span class="kw">then </span>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</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="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E3:27_1_1_3_1_4"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_1_3_1_4"><span class="lab"><font color="Green" title="E22">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_3_1_5"><span class="kw">suppose </span></a><a NAME="E1:27_1_1_3_1_5"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
( <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> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font></span>)</span></span><br/><div class="add"><a NAME="E2:27_1_1_3_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A10</font></span>: 
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></span>;<br/><a NAME="E3:27_1_1_3_1_5"/>
( the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_1_3_1_5"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:27_1_1_3_1_5"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <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></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_1_3_1_5"><span class="lab"><font color="Green" title="E23">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E4:27_1_1"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <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>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #); <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="metric_1.html#D3" target="_self" title="METRIC_1:def.3">METRIC_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="kw">thus </span><a NAME="E1:27_1_1_4"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</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="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:27_1_1"/>
(  <a href="metric_1.html#K6" title="METRIC_1:func.6">DiscreteSpace</a> <font color="Maroon" title="c1">A</font> is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>  &amp;  <a href="metric_1.html#K6" title="METRIC_1:func.6">DiscreteSpace</a> <font color="Maroon" title="c1">A</font> is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a>  &amp;  <a href="metric_1.html#K6" title="METRIC_1:func.6">DiscreteSpace</a> <font color="Maroon" title="c1">A</font> is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  &amp;  <a href="metric_1.html#K6" title="METRIC_1:func.6">DiscreteSpace</a> <font color="Maroon" title="c1">A</font> 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="#E1:27_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:27_1_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E3:27_1_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="metric_1.html#D7" target="_self" title="METRIC_1:def.7">Def7</a>, <a class="ref" href="metric_1.html#D8" target="_self" title="METRIC_1:def.8">Def8</a>, <a class="ref" href="metric_1.html#D9" target="_self" title="METRIC_1:def.9">Def9</a>, <a class="ref" href="metric_1.html#D10" target="_self" title="METRIC_1:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
