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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>,<a href="topmetr.html#K3">R^1</a> ;<br/>



<b>assume </b><a NAME="E1:96"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <a href="metric_1.html#K8">RealSpace</a> </span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">u1</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a> ;<br/><b>assume </b><a NAME="E1:96_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 ;<br/><b>set </b><font color="Maroon">s1</font> = <font color="Maroon">r</font>;<br/><a NAME="E2:96_1"/>
for <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span><br/> for <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>   st <font color="Olive">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">u</font>,<font color="Olive">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Maroon">u1</font>,<font color="Olive">w1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">w1</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a> ;<br/>



<b>assume </b><a NAME="E1:96_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">qw</font> = <font color="Maroon">w</font>, <font color="Maroon">qu</font> = <font color="Maroon">u</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E3:96_1_1"/><i><font color="Green">E58</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">u</font>,<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">qu</font> <a href="euclid.html#K20">-</a> <font color="Maroon">qw</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/>
<b>reconsider </b><font color="Maroon">tu</font> = <font color="Maroon">u1</font>, <font color="Maroon">tw</font> = <font color="Maroon">w1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<i><font color="Green">E59</font></i>:  <a href="metric_1.html#K4">dist</a> <font color="Maroon">u1</font>,<font color="Maroon">w1</font> = 
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#K8">RealSpace</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">u1</font>,<font color="Maroon">w1</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">tu</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">tw</font></span>)</span>
<b>by </b><i><a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>
;<br/>
<a NAME="E6:96_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">tu</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">qu</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E7:96_1_1"/>
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">qw</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E8:96_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">u1</font>,<font color="Maroon">w1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">qu</font> <a href="euclid.html#K20">-</a> <font color="Maroon">qw</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E9:96_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">u1</font>,<font color="Maroon">w1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T16" target="_self">Th16</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:96_1"/>
 ex <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span><br/> for <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>   st <font color="Olive">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">u</font>,<font color="Olive">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Maroon">u1</font>,<font color="Olive">w1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> ) )
 <b>by </b><i><a class="ref" href="jordan2c.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:96"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a>, <a class="ref" href="uniform1.html#T4">UNIFORM1:4</a></i>;<br/>


</div>
