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

<b>let </b><font color="Maroon">y</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">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font></span>)</span>
 ;<br/>

<a NAME="E3:16"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>  st <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:16_1"/>
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<a NAME="E2:16_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<b>consider </b><font color="Maroon">r0</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:16_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">r0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E6:16_1"/><i><font color="Green">E49</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">r</font>,<font color="Maroon">r0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="goboard6.html#T8">GOBOARD6:8</a></i>;<br/>
<a NAME="E7:16_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">y</font></span>)</span> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G1</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">r1</font>,<span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">y</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:16_1_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/><a NAME="E3:16_1_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><a NAME="E4:16_1_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font>
 <b>by </b><i>, <a class="ref" href="goboard6.html#T4">GOBOARD6:4</a></i>;<br/><a NAME="E5:16_1_1_1"/>
<font color="Maroon">G1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="goboard6.html#T6">GOBOARD6:6</a></i>;<br/><b>then consider </b><font color="Maroon">W1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:16_1_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E8:16_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">W1</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E9:16_1_1_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font>
 <b>by </b><i>, , <a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/><b>take </b>
<font color="Maroon">W1</font>
;<br/><b>thus </b><a NAME="E10:16_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E11:16_1_1_1"/>
<font color="Maroon">W1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/><a NAME="E12:16_1_1_1"/>
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">r</font>,<font color="Maroon">r0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_1_1_1_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:16_1_1_1_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E5:16_1_1_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E7:16_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
;<br/>
<a NAME="E8:16_1_1_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:16_1_1_1_1"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">ea1</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</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="E11:16_1_1_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">e1x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</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="E13:16_1_1_1_1"/><i><font color="Green">E104</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">r1</font>,<font color="Maroon">ea1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E14:16_1_1_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:16_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">r</font>,<font color="Maroon">e1x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E16:16_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">r</font>,<font color="Maroon">e1x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r0</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>hence </b><a NAME="E17:16_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">r</font>,<font color="Maroon">r0</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E13:16_1_1_1"/>
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>take </b><font color="Maroon">W</font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>;<br/><b>thus </b><a NAME="E2:16_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T13">PRE_TOPC:13</a></i>;<br/><b>thus </b><a NAME="E3:16_1_1_2"/>
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/><b>assume </b><a NAME="E4:16_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:16_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E7:16_1_1_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E8:16_1_1_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><i><font color="Green">E108</font></i>: <font color="Maroon">x</font> = 
<font color="Maroon">y</font> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/><font color="Maroon">r</font> = 
<font color="Maroon">y</font> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/><a NAME="E12:16_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">r</font>,<font color="Maroon">r0</font>
 <b>by </b><i>, , <a class="ref" href="goboard6.html#T4">GOBOARD6:4</a></i>;<br/><b>hence </b><a NAME="E13:16_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:16"/>
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>


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