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

<b>let </b><font color="Maroon">D</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> ;<br/>



<b>set </b><font color="Maroon">DF</font> =  <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #);<br/>
<a NAME="E1:45"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">f</font> <a href="pcomps_1.html#R1" target="_self">is_metric_of</a> <font color="Maroon">D</font> implies  <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) is   <a href="metric_1.html#NM1">MetrSpace</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">f</font> <a href="pcomps_1.html#R1" target="_self">is_metric_of</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:45_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) holds <br/> ( (  <a href="metric_1.html#K2">dist</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) &amp; ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> implies  <a href="metric_1.html#K2">dist</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> 0 ) &amp;  <a href="metric_1.html#K2">dist</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Olive">b</font>,<font color="Olive">a</font> &amp;  <a href="metric_1.html#K2">dist</a> <font color="Olive">a</font>,<font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Olive">b</font>,<font color="Olive">c</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #), <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #);<br/>

<a NAME="E1:45_1_1"/><i><font color="Green">E28</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:45_1_1"/>
(  <a href="metric_1.html#K2">dist</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="pcomps_1.html#T2" target="_self">Th2</a></i>;<br/>

<a NAME="E3:45_1_1"/>
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:45_1_1"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="pcomps_1.html#T2" target="_self">Th2</a></i>;<br/>

<a NAME="E5:45_1_1"/><i><font color="Green">E32</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E6:45_1_1"/>
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:45_1_1"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="pcomps_1.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:45_1"/>
 <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) is   <a href="metric_1.html#NM1">MetrSpace</a>
 <b>by </b><i><a class="ref" href="metric_1.html#T6">METRIC_1:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:45"/>
(  <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) is   <a href="metric_1.html#NM1">MetrSpace</a> implies <font color="Maroon">f</font> <a href="pcomps_1.html#R1" target="_self">is_metric_of</a> <font color="Maroon">D</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_2"/><i><font color="Green">E33</font></i>: 
 <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) is   <a href="metric_1.html#NM1">MetrSpace</a>
 ;<br/>

<a NAME="E2:45_2"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) holds <br/> ( ( the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) &amp; ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> implies the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> 0 ) &amp; the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Olive">b</font>,<font color="Olive">a</font> &amp; the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Olive">b</font>,<font color="Olive">c</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #), <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #);<br/>

<a NAME="E1:45_2_1"/><i><font color="Green">E48</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:45_2_1"/>
( the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="metric_1.html#T1">METRIC_1:1</a>, <a class="ref" href="metric_1.html#T2">METRIC_1:2</a></i>;<br/>

<a NAME="E3:45_2_1"/>
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:45_2_1"/>
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="metric_1.html#T3">METRIC_1:3</a></i>;<br/>

<a NAME="E5:45_2_1"/><i><font color="Green">E49</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E6:45_2_1"/>
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:45_2_1"/>
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:45_2"/>
<font color="Maroon">f</font> <a href="pcomps_1.html#R1" target="_self">is_metric_of</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="pcomps_1.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:45"/>
( <font color="Maroon">f</font> <a href="pcomps_1.html#R1" target="_self">is_metric_of</a> <font color="Maroon">D</font> iff  <a href="metric_1.html#G1">MetrStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">f</font> #) is   <a href="metric_1.html#NM1">MetrSpace</a> )
 <b>by </b><i/>;<br/>


</div>
