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

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">G</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">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> ;<br/>



<a NAME="E1:5"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">G</font> <a href="pcomps_1.html#R1">is_metric_of</a> <font color="Maroon">A</font> implies ( <font color="Maroon">G</font> is <a href="metric_1.html#V2">Reflexive</a> &amp; <font color="Maroon">G</font> is <a href="metric_1.html#V3">discerning</a> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Olive">b</font>,<font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</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"><font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">c</font></span>)</span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">G</font> <a href="pcomps_1.html#R1">is_metric_of</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:5_1"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">G</font> is <a href="metric_1.html#V2">Reflexive</a> &amp; <font color="Maroon">G</font> is <a href="metric_1.html#V3">discerning</a> &amp; <font color="Maroon">G</font> is <a href="metric_1.html#V4">symmetric</a> &amp; <font color="Maroon">G</font> is <a href="metric_1.html#V5">triangle</a> )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/><a NAME="E1:5_1_1"/>
<font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="metric_1.html#D6">METRIC_1:def 6</a></i>;<br/><b>hence </b><a NAME="E2:5_1_1"/>
<font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</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"><font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="metric_1.html#D5">METRIC_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:5_1"/>
( <font color="Maroon">G</font> is <a href="metric_1.html#V2">Reflexive</a> &amp; <font color="Maroon">G</font> is <a href="metric_1.html#V3">discerning</a> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Olive">b</font>,<font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</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"><font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">c</font></span>)</span> ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:5"/>
( <font color="Maroon">G</font> is <a href="metric_1.html#V2">Reflexive</a> &amp; <font color="Maroon">G</font> is <a href="metric_1.html#V3">discerning</a> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Olive">b</font>,<font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</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"><font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">c</font></span>)</span> ) implies <font color="Maroon">G</font> <a href="pcomps_1.html#R1">is_metric_of</a> <font color="Maroon">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:5_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">G</font> is <a href="metric_1.html#V2">Reflexive</a> &amp; <font color="Maroon">G</font> is <a href="metric_1.html#V3">discerning</a> )
 <b>and </b><br/><a NAME="E2:5_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Olive">b</font>,<font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</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"><font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">c</font></span>)</span>
 ;<br/>

<a NAME="E3:5_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Olive">b</font>,<font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Olive">c</font>,<font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E1:5_2_1"/>
<font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">c</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:5_2_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i>, <a class="ref" href="metric_1.html#D3">METRIC_1:def 3</a></i>;<br/>
<a NAME="E3:5_2_1"/>
<font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">b</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:5_2_1"/><b>then </b>
<font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i>, <a class="ref" href="metric_1.html#D3">METRIC_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:5_2_1"/>
<font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:5_2"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">G</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"><font color="Maroon">G</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"><font color="Maroon">G</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="5_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E1:5_2_2"/>
<font color="Maroon">G</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"><font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:5_2_2"/>
<font color="Maroon">G</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"><font color="Maroon">G</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"><font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:5_2"/><b>then </b>
( <font color="Maroon">G</font> is <a href="metric_1.html#V2">Reflexive</a> &amp; <font color="Maroon">G</font> is <a href="metric_1.html#V3">discerning</a> &amp; <font color="Maroon">G</font> is <a href="metric_1.html#V4">symmetric</a> &amp; <font color="Maroon">G</font> is <a href="metric_1.html#V5">triangle</a> )
 
<b>by </b><i>, , <a class="ref" href="metric_1.html#D5">METRIC_1:def 5</a>, <a class="ref" href="metric_1.html#D6">METRIC_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E6:5_2"/>
<font color="Maroon">G</font> <a href="pcomps_1.html#R1">is_metric_of</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:5"/>
( <font color="Maroon">G</font> <a href="pcomps_1.html#R1">is_metric_of</a> <font color="Maroon">A</font> iff ( <font color="Maroon">G</font> is <a href="metric_1.html#V2">Reflexive</a> &amp; <font color="Maroon">G</font> is <a href="metric_1.html#V3">discerning</a> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Olive">b</font>,<font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</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"><font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">c</font></span>)</span> ) ) )
 <b>by </b><i/>;<br/>


</div>
