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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being   <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">A</font>,<font color="Maroon" title="c1">A</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>  st <font color="Olive" title="b1">G</font> <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a> <font color="Maroon" title="c1">A</font> holds <br/> <a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">G</font>) <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a> <font color="Maroon" title="c1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</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">A</font>,<font color="Maroon" title="c1">A</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">G</font> <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a> <font color="Maroon" title="c1">A</font> implies  <a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>) <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a> <font color="Maroon" title="c1">A</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:7"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>) <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a> <font color="Maroon" title="c1">A</font></span><br/>

<a NAME="E2:7"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
 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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds  <span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <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"><span class="p2">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <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"><span class="p2">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <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="7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">a1</font> = <font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">a3</font> = <font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">a5</font> = <font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>);<br/>
<a NAME="E1:7_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
( <span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>)</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_6.html#D1" target="_self" title="METRIC_6:def.1">Def4</a></span>;<br/>
<a NAME="E2:7_1"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_6.html#D1" target="_self" title="METRIC_6:def.1">Def4</a></span>;<br/>
<a NAME="E3:7_1"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="metric_6.html#T2" target="_self" title="METRIC_6:th.2">Th2</a></span>;<br/>
<a NAME="E4:7_1"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="metric_6.html#T2" target="_self" title="METRIC_6:th.2">Th2</a></span>;<br/>
<a NAME="E5:7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A7</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></span>;<br/>
<span class="lab"><font color="Green" title="E12">A8</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7_1"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="xcmplx_1.html#T116" title="XCMPLX_1:th.116">XCMPLX_1:116</a></span>
<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" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E7:7_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="metric_6.html#T2" target="_self" title="METRIC_6:th.2">Th2</a></span>;<br/>
<a NAME="E8:7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A9</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>)</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>)</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_1"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E6:7_1"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="xcmplx_1.html#T130" title="XCMPLX_1:th.130">XCMPLX_1:130</a></span>;<br/>
<a NAME="E9:7_1"/><span class="lab"><font color="Green" title="E14">A10</font></span>: 
( <font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="metric_6.html#T2" target="_self" title="METRIC_6:th.2">Th2</a></span>;<br/>
<a NAME="E10:7_1"/>
<font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="pcomps_1.html#D6" title="PCOMPS_1:def.6">PCOMPS_1:def 6</a></span>;<br/>
<a NAME="E11:7_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T48" title="XREAL_1:th.48">XREAL_1:48</a></span>;<br/>
<a NAME="E12:7_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:7_1"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E3:7_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:7_1"/>
<span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E2:7_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E8:7_1"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="txt" href="#E9:7_1"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E3:7_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="xreal_1.html#T49" title="XREAL_1:th.49">XREAL_1:49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:7"/><span class="lab"><font color="Green" title="E7">A11</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <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>  iff <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )</span><br/>

<a NAME="E1:7_2"/><span class="lab"><font color="Green" title="E8">A12</font></span>: 
( <span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:7_2_1"/>
<span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font></span><br/>

<a NAME="E2:7_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_6.html#D1" target="_self" title="METRIC_6:def.1">Def4</a></span>;<br/>
<a NAME="E3:7_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="metric_6.html#T2" target="_self" title="METRIC_6:th.2">Th2</a></span>;<br/>
<a NAME="E4:7_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_2_1"><span class="lab"><font color="Green" title="E9">A13</font></span></a>, <a class="ref" href="xcmplx_1.html#T50" title="XCMPLX_1:th.50">XCMPLX_1:50</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:7_2_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="pcomps_1.html#D6" title="PCOMPS_1:def.6">PCOMPS_1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:7_2"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies <span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:7_2_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:7_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="pcomps_1.html#D6" title="PCOMPS_1:def.6">PCOMPS_1:def 6</a></span>;<br/>
<a NAME="E3:7_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E4:7_2_2"/>
<span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_6.html#D1" target="_self" title="METRIC_6:def.1">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:7_2"/>
( <span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2"><span class="lab"><font color="Green" title="E8">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:7"/>
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds  <span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <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> <span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <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>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>)</span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>)</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="pcomps_1.html#D6" title="PCOMPS_1:def.6">PCOMPS_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>)</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="pcomps_1.html#D6" title="PCOMPS_1:def.6">PCOMPS_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_6.html#D1" target="_self" title="METRIC_6:def.1">Def4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:7_3"/>
<span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_6.html#D1" target="_self" title="METRIC_6:def.1">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:7"/>
 <a href="metric_6.html#K1" title="METRIC_6:func.1">bounded_metric</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">G</font>) <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E7">A11</font></span></a>, <a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="pcomps_1.html#D6" title="PCOMPS_1:def.6">PCOMPS_1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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