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

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c4">X</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c5">f1</font>, <font color="Maroon" title="c6">f2</font>, <font color="Maroon" title="c7">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">X</font>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:17"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c5">f1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E2:17"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c6">f2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E3:17"/><i><font color="Green" title="E16">A3</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X</font> holds  <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1"/>
( ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 ) or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:17_1_1"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:17_1_1"/><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E3:17_1_1"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X</font><br/> for <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span>  st <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/> ex <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">X</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c7">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X</font>;<br/><b>let </b><font color="Maroon" title="c9">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:17_1_1_1"/>
( <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font> &amp; <font color="Maroon" title="c9">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 ;<br/>

<a NAME="E2:17_1_1_1"/><b>then </b><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</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="c9">V</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">r</font> = <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c3">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">r0</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="E5:17_1_1_1"/><i><font color="Green" title="E20">A7</font></i>: 
<font color="Maroon" title="c11">r0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>and </b><br/><a NAME="E6:17_1_1_1"/><i><font color="Green" title="E21">A8</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c11">r0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">V</font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E2:17_1_1_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="goboard6.html#T8" title="GOBOARD6:th.8">GOBOARD6:8</a></i>;<br/>
<a NAME="E7:17_1_1_1"/><i><font color="Green" title="E22">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font></span>)</span> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E8:17_1_1_1"/><i><font color="Green" title="E23">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">y</font></span>)</span> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E9:17_1_1_1"/><i><font color="Green" title="E24">A11</font></i>: 
<font color="Maroon" title="c11">r0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="topalg_1.html#E5:17_1_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">r1</font> = <font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c3">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">G1</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c12">r1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">r0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/>
<a NAME="E12:17_1_1_1"/><i><font color="Green" title="E25">A12</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:17_1_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></i>;<br/>
<a NAME="E13:17_1_1_1"/><i><font color="Green" title="E26">A13</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="topalg_1.html#E2:17_1_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></i>;<br/>
<a NAME="E14:17_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="topalg_1.html#E9:17_1_1_1"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="topalg_1.html#E12:17_1_1_1"><i><font color="Green" title="E25">A12</font></i></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></i>;<br/>
<a NAME="E15:17_1_1_1"/><b>then </b><i><font color="Green" title="E27">A14</font></i>: 
<font color="Maroon" title="c12">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">G1</font>
 
<b>by </b><i><a class="ref" href="goboard6.html#T4" title="GOBOARD6:th.4">GOBOARD6:4</a></i>;<br/>
<a NAME="E16:17_1_1_1"/>
<font color="Maroon" title="c13">G1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E7:17_1_1_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="goboard6.html#T6" title="GOBOARD6:th.6">GOBOARD6:6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">W1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">X</font><b> such that </b><br/><a NAME="E18:17_1_1_1"/><i><font color="Green" title="E28">A15</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W1</font>
 <b>and </b><br/><a NAME="E19:17_1_1_1"/><i><font color="Green" title="E29">A16</font></i>: 
<font color="Maroon" title="c14">W1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E20:17_1_1_1"/><i><font color="Green" title="E30">A17</font></i>: 
<font color="Maroon" title="c5">f1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c14">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">G1</font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E1:17"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="topalg_1.html#E15:17_1_1_1"><i><font color="Green" title="E27">A14</font></i></a>, <a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">r2</font> = <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c3">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">G2</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c15">r2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">r0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/>
<a NAME="E23:17_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="topalg_1.html#E9:17_1_1_1"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="topalg_1.html#E13:17_1_1_1"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></i>;<br/>
<a NAME="E24:17_1_1_1"/><b>then </b><i><font color="Green" title="E31">A18</font></i>: 
<font color="Maroon" title="c15">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">G2</font>
 
<b>by </b><i><a class="ref" href="goboard6.html#T4" title="GOBOARD6:th.4">GOBOARD6:4</a></i>;<br/>
<a NAME="E25:17_1_1_1"/>
<font color="Maroon" title="c16">G2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E8:17_1_1_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="goboard6.html#T6" title="GOBOARD6:th.6">GOBOARD6:6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">W2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">X</font><b> such that </b><br/><a NAME="E27:17_1_1_1"/><i><font color="Green" title="E32">A19</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">W2</font>
 <b>and </b><br/><a NAME="E28:17_1_1_1"/><i><font color="Green" title="E33">A20</font></i>: 
<font color="Maroon" title="c17">W2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E29:17_1_1_1"/><i><font color="Green" title="E34">A21</font></i>: 
<font color="Maroon" title="c6">f2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c17">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">G2</font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E2:17"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="topalg_1.html#E24:17_1_1_1"><i><font color="Green" title="E31">A18</font></i></a>, <a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<b>take </b><font color="Maroon" title="c18">W</font> = <font color="Maroon" title="c14">W1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c17">W2</font>;<br/>

<b>thus </b><a NAME="E30:17_1_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">W</font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E18:17_1_1_1"><i><font color="Green" title="E28">A15</font></i></a>, <a class="txt" href="topalg_1.html#E27:17_1_1_1"><i><font color="Green" title="E32">A19</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>

<b>thus </b><a NAME="E31:17_1_1_1"/>
<font color="Maroon" title="c18">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="topalg_1.html#E19:17_1_1_1"><i><font color="Green" title="E29">A16</font></i></a>, <a class="txt" href="topalg_1.html#E28:17_1_1_1"><i><font color="Green" title="E33">A20</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>;<br/>

<a NAME="E32:17_1_1_1"/>
<font color="Maroon" title="c7">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c18">W</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="c10">r</font>,<font color="Maroon" title="c11">r0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">a</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="E1:17_1_1_1_1"/>
<font color="Maroon" title="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c18">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c20">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:17_1_1_1_1"/><i><font color="Green" title="E35">A22</font></i>: 
<font color="Maroon" title="c20">z</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="c7">g</font>
 <b>and </b><br/><a NAME="E4:17_1_1_1_1"/><i><font color="Green" title="E36">A23</font></i>: 
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">W</font>
 <b>and </b><br/><a NAME="E5:17_1_1_1_1"/><i><font color="Green" title="E37">A24</font></i>: 
<font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">a</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/>
<a NAME="E6:17_1_1_1_1"/><i><font color="Green" title="E38">A25</font></i>: 
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W1</font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E4:17_1_1_1_1"><i><font color="Green" title="E36">A23</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">z</font> = <font color="Maroon" title="c20">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X</font> <b>by </b><i><a class="txt" href="topalg_1.html#E3:17_1_1_1_1"><i><font color="Green" title="E35">A22</font></i></a></i>;<br/>
<a NAME="E8:17_1_1_1_1"/><i><font color="Green" title="E39">A26</font></i>: 
<font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X</font>
 
;<br/>
<a NAME="E9:17_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c21">z</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="c5">f1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:17_1_1_1_1"/><b>then </b><i><font color="Green" title="E40">A27</font></i>: 
<font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c14">W1</font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E6:17_1_1_1_1"><i><font color="Green" title="E38">A25</font></i></a>, <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="c22">ea1</font> = <font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c3">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/>
<a NAME="E12:17_1_1_1_1"/><i><font color="Green" title="E41">A28</font></i>: 
<font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">W2</font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E4:17_1_1_1_1"><i><font color="Green" title="E36">A23</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E13:17_1_1_1_1"/>
<font color="Maroon" title="c21">z</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="c6">f2</font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E8:17_1_1_1_1"><i><font color="Green" title="E39">A26</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:17_1_1_1_1"/><b>then </b><i><font color="Green" title="E42">A29</font></i>: 
<font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c17">W2</font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E12:17_1_1_1_1"><i><font color="Green" title="E41">A28</font></i></a>, <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="c23">ea2</font> = <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c3">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/>
<a NAME="E16:17_1_1_1_1"/><i><font color="Green" title="E43">A30</font></i>: 
<font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">z</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">z</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E3:17"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="topalg_1.html#E5:17_1_1_1_1"><i><font color="Green" title="E37">A24</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c24">e1x</font> = <font color="Maroon" title="c19">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c3">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/>
<a NAME="E18:17_1_1_1_1"/><i><font color="Green" title="E44">A31</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c12">r1</font>,<font color="Maroon" title="c22">ea1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E20:17_1_1_1"><i><font color="Green" title="E30">A17</font></i></a>, <a class="txt" href="topalg_1.html#E10:17_1_1_1_1"><i><font color="Green" title="E40">A27</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E19:17_1_1_1_1"/><i><font color="Green" title="E45">A32</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c15">r2</font>,<font color="Maroon" title="c23">ea2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E29:17_1_1_1"><i><font color="Green" title="E34">A21</font></i></a>, <a class="txt" href="topalg_1.html#E14:17_1_1_1_1"><i><font color="Green" title="E42">A29</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E20:17_1_1_1_1"/>
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E3:17"><i><font color="Green" title="E16">A3</font></i></a></i>;<br/>
<a NAME="E21:17_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c24">e1x</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="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">r0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">r0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:17_1_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="topalg_1.html#E2:17_1_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="topalg_1.html#E16:17_1_1_1_1"><i><font color="Green" title="E43">A30</font></i></a>, <a class="txt" href="topalg_1.html#E18:17_1_1_1_1"><i><font color="Green" title="E44">A31</font></i></a>, <a class="txt" href="topalg_1.html#E19:17_1_1_1_1"><i><font color="Green" title="E45">A32</font></i></a>, <a class="ref" href="topalg_1.html#T14" target="_self" title="TOPALG_1:th.14">Th14</a></i>;<br/>
<a NAME="E22:17_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c24">e1x</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="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">r0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">r0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T48" title="XCMPLX_1:th.48">XCMPLX_1:48</a></i>;<br/>
<a NAME="E23:17_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c24">e1x</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="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">r0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">r0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T48" title="XCMPLX_1:th.48">XCMPLX_1:48</a></i>;<br/>
<a NAME="E24:17_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c24">e1x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r0</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"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">r0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E12:17_1_1_1"><i><font color="Green" title="E25">A12</font></i></a>, <a class="ref" href="xcmplx_1.html#T98" title="XCMPLX_1:th.98">XCMPLX_1:98</a></i>;<br/>
<a NAME="E25:17_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c24">e1x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r0</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="c11">r0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E13:17_1_1_1"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="xcmplx_1.html#T98" title="XCMPLX_1:th.98">XCMPLX_1:98</a></i>;<br/>
<b>hence </b><a NAME="E26:17_1_1_1_1"/>
<font color="Maroon" title="c19">a</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="c10">r</font>,<font color="Maroon" title="c11">r0</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>
<b>hence </b><a NAME="E33:17_1_1_1"/>
<font color="Maroon" title="c7">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c18">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">V</font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E6:17_1_1_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:17_1_1"/>
<font color="Maroon" title="c7">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2"><b>suppose </b></a><a NAME="E1:17_1_2"/><i><font color="Green" title="E17">A33</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_2"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X</font> holds  <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X</font>;<br/>

<b>thus </b><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E3:17"><i><font color="Green" title="E16">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E1:17_1_2"><i><font color="Green" title="E17">A33</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon" title="c2">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_2"/>
<font color="Maroon" title="c7">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_1.html#E2:17"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="topalg_1.html#T16" target="_self" title="TOPALG_1:th.16">Th16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_3"><b>suppose </b></a><a NAME="E1:17_1_3"/><i><font color="Green" title="E17">A34</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_3"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X</font> holds  <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X</font>;<br/>

<b>thus </b><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E3:17"><i><font color="Green" title="E16">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c3">n</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E1:17_1_3"><i><font color="Green" title="E17">A34</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_3"/>
<font color="Maroon" title="c7">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_1.html#E1:17"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="topalg_1.html#T16" target="_self" title="TOPALG_1:th.16">Th16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
