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

<b>set </b><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"/>
(  <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#V6" title="METRIC_1:attr.6">Reflexive</a> &amp;  <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#V7" title="METRIC_1:attr.7">discerning</a> &amp;  <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#V8" title="METRIC_1:attr.8">symmetric</a> &amp;  <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#V9" title="METRIC_1:attr.9">triangle</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><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> #) is <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D7" target="_self" title="METRIC_1:def.7">METRIC_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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#V7" title="METRIC_1:attr.7">discerning</a> &amp;  <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#V8" title="METRIC_1:attr.8">symmetric</a> &amp;  <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#V9" title="METRIC_1:attr.9">triangle</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><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> #); <i><font color="Red">:: according to </font></i><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:()"><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 <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/>

<b>thus </b><a NAME="E1:27_1_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="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</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>

<b>thus </b><a NAME="E2:27_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> #) is <a href="metric_1.html#V3" title="METRIC_1:attr.3">discerning</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D8" target="_self" title="METRIC_1:def.8">METRIC_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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#V8" title="METRIC_1:attr.8">symmetric</a> &amp;  <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#V9" title="METRIC_1:attr.9">triangle</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_2"><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 <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> #); <i><font color="Red">:: according to </font></i><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:()"><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 <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/>

<b>assume </b><a NAME="E1:27_1_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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>hence </b><a NAME="E2:27_1_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>
 <b>by </b><i><a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</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>

<b>thus </b><a NAME="E3:27_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> #) is <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D9" target="_self" title="METRIC_1:def.9">METRIC_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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#V9" title="METRIC_1:attr.9">triangle</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3"><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 <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> #); <i><font color="Red">:: according to </font></i><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:()"><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 <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_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1_3_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_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_1_1_1"/><i><font color="Green" title="E18">A1</font></i>: 
<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:()"><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 <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"><b>hence </b>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
<b>by </b><i><a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></i>
<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>
<b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></i>
;<br/> <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_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:()"><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 <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"><b>hence </b><a NAME="E2:27_1_1_1_3_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_1_1_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="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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:27_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> #) is <a href="metric_1.html#V5" title="METRIC_1:attr.5">triangle</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D10" target="_self" title="METRIC_1:def.10">METRIC_1:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><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> #); <i><font color="Red">:: according to </font></i><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:()"><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 <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_1_4"/><i><font color="Green" title="E18">A2</font></i>: 
( 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> )
 
<b>by </b><i><a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1_4_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_1_4_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_4_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:()"><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 <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"><b>hence </b><a NAME="E2:27_1_1_1_4_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>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_4"><i><font color="Green" title="E18">A2</font></i></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_4_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:()"><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 <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"><b>hence </b><a NAME="E2:27_1_1_1_4_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>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_4"><i><font color="Green" title="E18">A2</font></i></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_4_1_3"><b>suppose </b></a><a NAME="E1:27_1_1_1_4_1_3"/><i><font color="Green" title="E19">A3</font></i>: 
( <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:()"><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 <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_1_4_1_3"/><b>then </b><i><font color="Green" title="E20">A4</font></i>: 
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> 
 <b>by </b><i><a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></i>;<br/><a NAME="E3:27_1_1_1_4_1_3"/><i><font color="Green" title="E21">A5</font></i>: 
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
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_4_1_3"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></i>;<br/><a NAME="E4:27_1_1_1_4_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="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_4_1_3"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></i>;<br/><b>hence </b><a NAME="E5:27_1_1_1_4_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>
 <b>by </b><i><a class="txt" href="#E2:27_1_1_1_4_1_3"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E3:27_1_1_1_4_1_3"><i><font color="Green" title="E21">A5</font></i></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_4_1_4"><b>suppose </b></a><a NAME="E1:27_1_1_1_4_1_4"/><i><font color="Green" title="E19">A6</font></i>: 
( <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:()"><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 <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_1_4_1_4"/><b>then </b>
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> 
 <b>by </b><i><a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></i>;<br/><b>hence </b><a NAME="E3:27_1_1_1_4_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>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_4_1_4"><i><font color="Green" title="E19">A6</font></i></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_4_1_5"><b>suppose </b></a><a NAME="E1:27_1_1_1_4_1_5"/>
( <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:()"><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 <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_1_4_1_5"/><b>then </b>
( 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 &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="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="metric_1.html#D11" target="_self" title="METRIC_1:def.11">Def11</a></i>;<br/><b>hence </b><a NAME="E3:27_1_1_1_4_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/><b>thus </b><a NAME="E4:27_1_1_1_4_1_5"/>
verum
 ; <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></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E5:27_1_1_1"/>
verum
 ; <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>
<b>hence </b><a NAME="E2:27_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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
