<?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/>



<b>assume </b><a NAME="E1:7"/><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>
 ;<br/>

<a NAME="E2:7"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds <br/> ( <span class="p1">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> 0 iff <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><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>;<br/>

<a NAME="E1:7_1"/><i><font color="Green">E40</font></i>: 
( <span class="p1">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_1"/>
<span class="p1">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:7_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<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#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<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></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:7_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:7_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <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> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:7_1_1"/><b>then </b>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 <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">b</font></span>)</span>
 
;<br/>
<a NAME="E6:7_1_1"/><b>then </b>
<font color="Maroon">G</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
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T50">XCMPLX_1:50</a></i>;<br/>
<b>hence </b><a NAME="E7:7_1_1"/>
<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#D7">PCOMPS_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:7_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> implies <span class="p1">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:7_1_2"/><b>then </b>
<font color="Maroon">G</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
 
<b>by </b><i>, <a class="ref" href="pcomps_1.html#D7">PCOMPS_1:def 7</a></i>;<br/>
<a NAME="E3:7_1_2"/><b>then </b>
<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#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<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></span>)</span> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<b>hence </b><a NAME="E4:7_1_2"/>
<span class="p1">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_1"/>
( <span class="p1">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <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/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:7"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <span class="p1">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Olive">b</font>,<font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_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>;<br/>

<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#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<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></span>)</span> = 
<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#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<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></span>)</span>
<b>by </b><i>, <a class="ref" href="pcomps_1.html#D7">PCOMPS_1:def 7</a></i>
<br/>.= 
<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#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<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></span>)</span>
<b>by </b><i>, <a class="ref" href="pcomps_1.html#D7">PCOMPS_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:7_2"/>
<span class="p1">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:7"/>
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  <span class="p1">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <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="7_3"><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="E1:7_3"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</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#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<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></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:7_3"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</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> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<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></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:7_3"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</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#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<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></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:7_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:7_3"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <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> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:7_3"/><b>then </b><i><font color="Green">E53</font></i>: 
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> 1
 
;<br/>
<a NAME="E7:7_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:7_3"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <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> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:7_3"/><b>then </b><i><font color="Green">E54</font></i>: 
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 <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>
 
;<br/>
<a NAME="E10:7_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:7_3"/><b>then </b><i><font color="Green">E55</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <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">c</font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>set </b><font color="Maroon">a1</font> = <font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">a3</font> = <font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>;<br/>
<b>set </b><font color="Maroon">a5</font> = <font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>;<br/>
<a NAME="E12:7_3"/><i><font color="Green">E59</font></i>: 
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 <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/>;<br/>
<a NAME="E13:7_3"/><i><font color="Green">E60</font></i>: 
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<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></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<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#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<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></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<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#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<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></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4">(<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></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<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#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4">(<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></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<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></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T117">XCMPLX_1:117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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#K4">*</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<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#K4">*</a> <span class="p4">(<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></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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#K4">*</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<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#K4">*</a> <span class="p4">(<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></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<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></span>)</span></span>)</span>

;<br/>
<a NAME="E15:7_3"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4">(<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></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<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#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4">(<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></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<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></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><span class="p0">(<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#K4">*</a> <span class="p0">(<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></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><span class="p0">(<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#K4">*</a> <span class="p0">(<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></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4">(<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></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<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> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p5">(<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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p5">(<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></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4">(<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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4">(<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></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<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></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T131">XCMPLX_1:131</a></i>;<br/>
<a NAME="E16:7_3"/>
<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>, <a class="ref" href="pcomps_1.html#D7">PCOMPS_1:def 7</a></i>;<br/>
<a NAME="E17:7_3"/><b>then </b><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<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="p2">(<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></span>)</span> <a href="real_1.html#K5">-</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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E18:7_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">G</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E19:7_3"/><i><font color="Green">E64</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#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E20:7_3"/><i><font color="Green">E65</font></i>: 
<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#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E21:7_3"/><i><font color="Green">E66</font></i>: 
<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#K4">*</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="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>
<a NAME="E22:7_3"/><i><font color="Green">E67</font></i>: 
<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#K4">*</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="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>
<a NAME="E23:7_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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#K4">*</a> <span class="p2">(<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></span>)</span> <a href="real_1.html#K4">*</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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>
<a NAME="E24:7_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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#K4">*</a> <span class="p2">(<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></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K4">*</a> <span class="p3">(<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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<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></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E25:7_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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#K4">*</a> <span class="p2">(<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></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K4">*</a> <span class="p3">(<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></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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#K4">*</a> <span class="p4">(<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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<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></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E26:7_3"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<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></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K4">*</a> <span class="p3">(<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></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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#K4">*</a> <span class="p4">(<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></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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#K4">*</a> <span class="p5">(<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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<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></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E27:7_3"/>
0 <a href="nat_1.html#K1">+</a> 1 <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> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E28:7_3"/><b>then </b><i><font color="Green">E69</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <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">b</font></span>)</span>
 
;<br/>
<a NAME="E29:7_3"/>
0 <a href="nat_1.html#K1">+</a> 1 <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">c</font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E30:7_3"/><b>then </b><i><font color="Green">E70</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <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>
 
;<br/>
<a NAME="E31:7_3"/>
0 <a href="nat_1.html#K1">+</a> 1 <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> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E32:7_3"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <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>
 
;<br/>
<a NAME="E33:7_3"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<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></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/>
<a NAME="E34:7_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<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></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<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></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/>
<a NAME="E35:7_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4">(<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></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<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#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4">(<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></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<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></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T125">REAL_2:125</a></i>;<br/>
<b>hence </b><a NAME="E36:7_3"/>
<span class="p1">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <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="real_2.html#T105">REAL_2:105</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:7"/>
 <a href="metric_6.html#K1" target="_self">bounded_metric</a> <font color="Maroon">A</font>,<font color="Maroon">G</font> <a href="pcomps_1.html#R1">is_metric_of</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="pcomps_1.html#D7">PCOMPS_1:def 7</a></i>;<br/>


</div>
