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

<b>let </b><font color="Maroon" title="c1">D</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c1">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">DF</font> =  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">f</font> #);<br/>
<a NAME="E1:45"/><i><font color="Green" title="E33">A1</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a> <font color="Maroon" title="c1">D</font> implies  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">f</font> #) is   <a href="metric_1.html#NM1" title="METRIC_1:NM.1">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" title="E34">A2</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a> <font color="Maroon" title="c1">D</font>
 ;<br/>

<a NAME="E2:45_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">f</font> #) holds <br/> ( (  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">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="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) &amp; ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> implies  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">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" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">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">D</font>,<font color="Maroon" title="c2">f</font> #);<br/>

<a NAME="E1:45_1_1"/><i><font color="Green" title="E35">A3</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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:45_1_1"/>
(  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="txt" href="pcomps_1.html#E1:45_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="pcomps_1.html#D7" target="_self" title="PCOMPS_1:def.7">Def7</a></i>;<br/>

<a NAME="E3:45_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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:45_1_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E1:45_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="pcomps_1.html#E1:45_1_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="pcomps_1.html#D7" target="_self" title="PCOMPS_1:def.7">Def7</a></i>;<br/>

<a NAME="E5:45_1_1"/><i><font color="Green" title="E36">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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E6:45_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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:45_1_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E1:45_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="pcomps_1.html#E1:45_1_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="pcomps_1.html#E5:45_1_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="pcomps_1.html#D7" target="_self" title="PCOMPS_1:def.7">Def7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:45_1"/>
 <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">f</font> #) is   <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>
 <b>by </b><i><a class="ref" href="metric_1.html#T6" title="METRIC_1:th.6">METRIC_1:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:45"/>
(  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">f</font> #) is   <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a> implies <font color="Maroon" title="c2">f</font> <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a> <font color="Maroon" title="c1">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" title="E34">A5</font></i>: 
 <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">f</font> #) is   <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>
 ;<br/>

<a NAME="E2:45_2"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">f</font> #) holds <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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">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="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) &amp; ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> implies 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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> &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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">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" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">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">D</font>,<font color="Maroon" title="c2">f</font> #);<br/>

<a NAME="E1:45_2_1"/><i><font color="Green" title="E35">A6</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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:45_2_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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="txt" href="pcomps_1.html#E1:45_2"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a>, <a class="ref" href="metric_1.html#T2" title="METRIC_1:th.2">METRIC_1:2</a></i>;<br/>

<a NAME="E3:45_2_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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:45_2_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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E1:45_2"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="pcomps_1.html#E1:45_2_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="metric_1.html#T3" title="METRIC_1:th.3">METRIC_1:3</a></i>;<br/>

<a NAME="E5:45_2_1"/><i><font color="Green" title="E36">A7</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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E6:45_2_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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:45_2_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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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">D</font>,<font color="Maroon" title="c2">f</font> #) <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E1:45_2"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="pcomps_1.html#E1:45_2_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="pcomps_1.html#E5:45_2_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:45_2"/>
<font color="Maroon" title="c2">f</font> <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D7" target="_self" title="PCOMPS_1:def.7">Def7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:45"/>
( <font color="Maroon" title="c2">f</font> <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a> <font color="Maroon" title="c1">D</font> iff  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">f</font> #) is   <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a> )
 <b>by </b><i><a class="txt" href="pcomps_1.html#E1:45"><i><font color="Green" title="E33">A1</font></i></a></i>;<br/>


</div>
