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

<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/>

<b>given </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><a NAME="E1:41"/><i><font color="Green" title="E26">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">b</font>
 ;<br/>

<a NAME="E2:41"/>
for <font color="Olive" title="b1">W</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <br/> for <font color="Olive" title="b2">G</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">W</font>  ex <font color="Olive" title="b3">H</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">W</font> st <font color="Maroon" title="c1">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b3">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">W</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/><b>let </b><font color="Maroon" title="c5">G</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">W</font>;<br/>



<a NAME="E1:41_1"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E3:41_1"/><i><font color="Green" title="E28">A3</font></i>: 
( <font color="Maroon" title="c6">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q</font> )
 <b>by </b><i><a class="ref" href="tops_1.html#T54" title="TOPS_1:th.54">TOPS_1:54</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">Y</font> = <font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">W</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" target="_self" title="TOPMETR:th.24">Th24</a>, <a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E6:41_1"/><i><font color="Green" title="E29">A4</font></i>: 
( <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">Y</font>,<font color="Maroon" title="c8">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Q</font> )
 <b>by </b><i><a class="txt" href="topmetr.html#E3:41_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="topmetr.html#T22" target="_self" title="TOPMETR:th.22">Th22</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_1_1"/><i><font color="Green" title="E30">A5</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c9">H</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/><a NAME="E2:41_1_1_1_1"/>
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c4">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  )
 ;<br/><a NAME="E3:41_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">H</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">W</font> <b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/><a NAME="E5:41_1_1_1_1"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> iff  ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">H</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:41_1_1_1_1_1"/>
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">H</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_1_1_1_1_1"/><i><font color="Green" title="E31">A6</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<b>take </b>
0
;<br/>

<a NAME="E2:41_1_1_1_1_1_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="topmetr.html#T24" target="_self" title="TOPMETR:th.24">Th24</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:41_1_1_1_1_1_1"/>
( 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> &amp; 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">H</font> )
 ;<br/>

<b>thus </b><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 = 
<span class="p1">(<span class="default">0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 0</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="topmetr.html#E1:41"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="topmetr.html#E1:41_1_1_1_1"><i><font color="Green" title="E30">A5</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">y</font>
<b>by </b><i><a class="txt" href="topmetr.html#E1:41_1_1_1_1_1_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:41_1_1_1_1_1"/><i><font color="Green" title="E31">A7</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">H</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c12">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="topmetr.html#E2:41_1_1_1_1_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="topmetr.html#T24" target="_self" title="TOPMETR:th.24">Th24</a></i>;<br/>
<font color="Maroon" title="c11">y</font> = 
<span class="p1">(<span class="default">0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="topmetr.html#E1:41"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="topmetr.html#E1:41_1_1_1_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="topmetr.html#E2:41_1_1_1_1_1"><i><font color="Green" title="E31">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font>

;<br/>
<b>hence </b><a NAME="E5:41_1_1_1_1_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:41_1_1_1_1"/><b>then </b><i><font color="Green" title="E31">A8</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:41_1_1_1_1"/><i><font color="Green" title="E32">A9</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">W'</font> = <font color="Maroon" title="c4">W</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="topmetr.html#T24" target="_self" title="TOPMETR:th.24">Th24</a></i>;<br/><font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">W</font> = 
<span class="p1">(<span class="default">0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c11">W'</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="topmetr.html#E1:41"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="topmetr.html#E1:41_1_1_1_1"><i><font color="Green" title="E30">A5</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font>

;<br/><a NAME="E10:41_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="topmetr.html#E1:41_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="topmetr.html#E6:41_1_1_1_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="topmetr.html#E7:41_1_1_1_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E11:41_1_1_1_1"/>
 ex <font color="Olive" title="b1">H</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">W</font> st <font color="Maroon" title="c1">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_1_2"/><i><font color="Green" title="E30">A10</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c9">d</font> = <font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>;<br/><a NAME="E2:41_1_1_1_2"/>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="topmetr.html#E1:41_1_1_1_2"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></i>;<br/><a NAME="E3:41_1_1_1_2"/><b>then </b>
2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a></i>;<br/><a NAME="E4:41_1_1_1_2"/><b>then </b><i><font color="Green" title="E31">A11</font></i>: 
<font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="topmetr.html#E6:41_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">W'</font> = <font color="Maroon" title="c4">W</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" target="_self" title="TOPMETR:th.24">Th24</a>, <a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">H</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c10">W'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" target="_self" title="TOPMETR:th.24">Th24</a>, <a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/><a NAME="E7:41_1_1_1_2"/>
<font color="Maroon" title="c11">H</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T21" target="_self" title="TOPMETR:th.21">Th21</a></i>;<br/><a NAME="E8:41_1_1_1_2"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c11">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">H</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/><a NAME="E9:41_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c11">H</font>
 <b>by </b><i><a class="txt" href="topmetr.html#E4:41_1_1_1_2"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="topmetr.html#E11"><i><font color="Green" title="E5">Lm1</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">H</font> = <font color="Maroon" title="c11">H</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">W</font> <b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/><a NAME="E11:41_1_1_1_2"/><i><font color="Green" title="E32">A12</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c12">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">Y</font>,<font color="Maroon" title="c8">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:41_1_1_1_2_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c12">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:41_1_1_1_2_1"/><i><font color="Green" title="E33">A13</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">H</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">y'</font> = <font color="Maroon" title="c13">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="topmetr.html#E3:41_1_1_1_2_1"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="topmetr.html#T24" target="_self" title="TOPMETR:th.24">Th24</a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">x'</font> = <font color="Maroon" title="c14">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="txt" href="topmetr.html#E3:41_1_1_1_2_1"><i><font color="Green" title="E33">A13</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">Y'</font> = <font color="Maroon" title="c7">Y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">W''</font> = <font color="Maroon" title="c10">W'</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="topmetr.html#T24" target="_self" title="TOPMETR:th.24">Th24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">x''</font> = <font color="Maroon" title="c16">x'</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E9:41_1_1_1_2_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">W'</font>,<font color="Maroon" title="c16">x'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="topmetr.html#E3:41_1_1_1_2_1"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E10:41_1_1_1_2_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">W''</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c19">x''</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="topmetr.html#E1:41_1_1_1_2"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="topmetr.html#T15" target="_self" title="TOPMETR:th.15">Th15</a>, <a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></i>;<br/>
<a NAME="E11:41_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">W''</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c19">x''</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E12:41_1_1_1_2_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">W''</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c19">x''</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>;<br/>
<a NAME="E13:41_1_1_1_2_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c18">W''</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c19">x''</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E14:41_1_1_1_2_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">Y'</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c19">x''</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="topmetr.html#E1:41"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<a NAME="E15:41_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E34">A14</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">Y'</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c15">y'</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="topmetr.html#E1:41"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="topmetr.html#E3:41_1_1_1_2_1"><i><font color="Green" title="E33">A13</font></i></a></i>;<br/>
<a NAME="E16:41_1_1_1_2_1"/>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="topmetr.html#E1:41_1_1_1_2"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>;<br/>
<a NAME="E17:41_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E35">A15</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T93" title="XCMPLX_1:th.93">XCMPLX_1:93</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">yy</font> = <font color="Maroon" title="c15">y'</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E19:41_1_1_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 
<b>by </b><i><a class="txt" href="topmetr.html#E6:41_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="real_2.html#T125" title="REAL_2:th.125">REAL_2:125</a></i>;<br/>
<a NAME="E20:41_1_1_1_2_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">Y'</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c15">y'</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font>
 
<b>by </b><i><a class="txt" href="topmetr.html#E15:41_1_1_1_2_1"><i><font color="Green" title="E34">A14</font></i></a>, <a class="txt" href="topmetr.html#E17:41_1_1_1_2_1"><i><font color="Green" title="E35">A15</font></i></a>, <a class="txt" href="topmetr.html#E38"><i><font color="Green" title="E25">Lm3</font></i></a></i>;<br/>
<a NAME="E21:41_1_1_1_2_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">Y</font>,<font color="Maroon" title="c20">yy</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15" target="_self" title="TOPMETR:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E22:41_1_1_1_2_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">Y</font>,<font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:41_1_1_1_2"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">Y</font>,<font color="Maroon" title="c8">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="topmetr.html#E3:41_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="topmetr.html#E6:41_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E13:41_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c1">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c12">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="topmetr.html#E11:41_1_1_1_2"><i><font color="Green" title="E32">A12</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E14:41_1_1_1_2"/>
 ex <font color="Olive" title="b1">H</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">W</font> st <font color="Maroon" title="c1">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:41_1"/>
 ex <font color="Olive" title="b1">H</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">W</font> st <font color="Maroon" title="c1">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:41"/>
<font color="Maroon" title="c1">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D3" title="BORSUK_1:def.3">BORSUK_1:def 3</a></i>;<br/>


</div>
