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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>assume </b><a NAME="E1:43"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">F</font> be   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>;<br/>

<b>assume </b><a NAME="E2:43"/><i><font color="Green" title="E24">A2</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> holds  <font color="Maroon" title="c3">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">f'</font> = <font color="Maroon" title="c2">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">F'</font> = <font color="Maroon" title="c3">F</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E5:43"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c4">f'</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E1:43"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">cT</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<br/>
<b>set </b><font color="Maroon" title="c7">TT</font> = <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">tt</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>;<br/><a NAME="E1:43_1"/>
<font color="Maroon" title="c8">tt</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 <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 ;<br/><a NAME="E2:43_1"/><b>then </b>
<font color="Maroon" title="c8">tt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">t1</font>, <font color="Maroon" title="c10">t2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:43_1"/><i><font color="Green" title="E26">A4</font></i>: 
( <font color="Maroon" title="c9">t1</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="c1">T</font> &amp; <font color="Maroon" title="c10">t2</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="c1">T</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">t1</font>,<font color="Maroon" title="c10">t2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">tt</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">t1</font> = <font color="Maroon" title="c9">t1</font>, <font color="Maroon" title="c12">t2</font> = <font color="Maroon" title="c10">t2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="nagata_1.html#E4:43_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/><a NAME="E6:43_1"/>
for <font color="Olive" title="b1">R</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>   st <font color="Olive" title="b1">R</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c5">F'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">R</font> holds <br/> ex <font color="Olive" title="b2">H</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Olive" title="b2">H</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">tt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H</font> &amp; <font color="Maroon" title="c5">F'</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">R</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">R</font> be   <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> ;<br/>

<a NAME="E1:43_1_1"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">t1</font>,<font color="Maroon" title="c12">t2</font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E4:43_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:43_1_1"/><i><font color="Green" title="E28">A6</font></i>: 
( <font color="Maroon" title="c13">R</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c5">F'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">R</font> )
 ;<br/>

<a NAME="E3:43_1_1"/><b>then </b><i><font color="Green" title="E29">A7</font></i>: 
( <font color="Maroon" title="c13">R</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">R</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</font> )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E2:43"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="nagata_1.html#E1:43_1_1"><i><font color="Green" title="E27">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">Ftt</font> = <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</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/>
<b>reconsider </b><font color="Maroon" title="c15">ft1</font> = <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t1</font>, <font color="Maroon" title="c16">ft2</font> = <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t2</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/>
<b>consider </b><font color="Maroon" title="c17">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="E7:43_1_1"/><i><font color="Green" title="E30">A8</font></i>: 
( <font color="Maroon" title="c17">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c14">Ftt</font>,<font color="Maroon" title="c17">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">R</font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E2:43_1_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">r'</font> = <font color="Maroon" title="c17">r</font> as   <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="E9:43_1_1"/><i><font color="Green" title="E31">A9</font></i>: 
<font color="Maroon" title="c18">r'</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="nagata_1.html#E7:43_1_1"><i><font color="Green" title="E30">A8</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="c19">A</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c15">ft1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">r'</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>, <font color="Maroon" title="c20">B</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c16">ft2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">r'</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</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="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E11:43_1_1"/><i><font color="Green" title="E32">A10</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">A</font> &amp; <font color="Maroon" title="c19">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c19">A</font> is   <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>  &amp; <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">B</font> &amp; <font color="Maroon" title="c20">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E9:43_1_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="nagata_1.html#E33"><i><font color="Green" title="E21">Lm7</font></i></a>, <a class="ref" href="topmetr.html#T21" title="TOPMETR:th.21">TOPMETR:21</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E12:43_1_1"/>
<font color="Maroon" title="c4">f'</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c11">t1</font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E5:43"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="tmap_1.html#T55" title="TMAP_1:th.55">TMAP_1:55</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">T1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E14:43_1_1"/><i><font color="Green" title="E33">A11</font></i>: 
( <font color="Maroon" title="c21">T1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c11">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">T1</font> &amp; <font color="Maroon" title="c4">f'</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c21">T1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">A</font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E11:43_1_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="tmap_1.html#T48" title="TMAP_1:th.48">TMAP_1:48</a></i>;<br/>
<a NAME="E15:43_1_1"/>
<font color="Maroon" title="c4">f'</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c12">t2</font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E5:43"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="tmap_1.html#T55" title="TMAP_1:th.55">TMAP_1:55</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">T2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E17:43_1_1"/><i><font color="Green" title="E34">A12</font></i>: 
( <font color="Maroon" title="c22">T2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c12">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">T2</font> &amp; <font color="Maroon" title="c4">f'</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c22">T2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">B</font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E11:43_1_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="tmap_1.html#T48" title="TMAP_1:th.48">TMAP_1:48</a></i>;<br/>
<a NAME="E18:43_1_1"/><i><font color="Green" title="E35">A13</font></i>: 
( <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">T1</font>,<font color="Maroon" title="c22">T2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">tt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">T1</font>,<font color="Maroon" title="c22">T2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E4:43_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="nagata_1.html#E14:43_1_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="txt" href="nagata_1.html#E17:43_1_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E19:43_1_1"/>
<font color="Maroon" title="c3">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">T1</font>,<font color="Maroon" title="c22">T2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">Fxy</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="E1:43_1_1_1"/>
<font color="Maroon" title="c23">Fxy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">T1</font>,<font color="Maroon" title="c22">T2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c24">xy</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:43_1_1_1"/><i><font color="Green" title="E36">A14</font></i>: 
( <font color="Maroon" title="c24">xy</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="c3">F</font> &amp; <font color="Maroon" title="c24">xy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">T1</font>,<font color="Maroon" title="c22">T2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <font color="Maroon" title="c23">Fxy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">xy</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>consider </b><font color="Maroon" title="c25">x</font>, <font color="Maroon" title="c26">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:43_1_1_1"/><i><font color="Green" title="E37">A15</font></i>: 
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">T1</font> &amp; <font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">T2</font> &amp; <font color="Maroon" title="c24">xy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c26">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E3:43_1_1_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">x</font> = <font color="Maroon" title="c25">x</font>, <font color="Maroon" title="c28">y</font> = <font color="Maroon" title="c26">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="nagata_1.html#E5:43_1_1_1"><i><font color="Green" title="E37">A15</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c29">Fxy1</font> = <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c27">x</font>,<font color="Maroon" title="c28">y</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> 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="E8:43_1_1_1"/><i><font color="Green" title="E38">A16</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c27">x</font>,<font color="Maroon" title="c28">y</font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E2:43"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>
<a NAME="E9:43_1_1_1"/>
( <font color="Maroon" title="c27">x</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="c1">T</font> &amp; <font color="Maroon" title="c28">y</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="c1">T</font> )
 
;<br/>
<a NAME="E10:43_1_1_1"/><b>then </b>
( <font color="Maroon" title="c27">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="c2">f</font> &amp; <font color="Maroon" title="c28">y</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="c2">f</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="E11:43_1_1_1"/><b>then </b><i><font color="Green" title="E39">A17</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c21">T1</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c22">T2</font> )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E5:43_1_1_1"><i><font color="Green" title="E37">A15</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="c30">fx</font> = <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x</font>, <font color="Maroon" title="c31">fy</font> = <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">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="E13:43_1_1_1"/>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c30">fx</font>,<font color="Maroon" title="c15">ft1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">r'</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c31">fy</font>,<font color="Maroon" title="c16">ft2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">r'</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E14:43_1_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="txt" href="nagata_1.html#E17:43_1_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="nagata_1.html#E11:43_1_1_1"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E14:43_1_1_1"/><b>then </b><i><font color="Green" title="E40">A18</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c30">fx</font>,<font color="Maroon" title="c15">ft1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c31">fy</font>,<font color="Maroon" title="c16">ft2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">r'</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">r'</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E15:43_1_1_1"/>
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c27">x</font>,<font color="Maroon" title="c28">y</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">y</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E3:43_1_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="nagata_1.html#E8:43_1_1_1"><i><font color="Green" title="E38">A16</font></i></a>, <a class="txt" href="nagata_1.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<a NAME="E16:43_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c27">x</font>,<font color="Maroon" title="c28">y</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c16">ft2</font>,<font color="Maroon" title="c31">fy</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15" title="TOPMETR:th.15">TOPMETR:15</a></i>;<br/>
<a NAME="E17:43_1_1_1"/><b>then </b><i><font color="Green" title="E41">A19</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c27">x</font>,<font color="Maroon" title="c28">y</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c30">fx</font>,<font color="Maroon" title="c15">ft1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c16">ft2</font>,<font color="Maroon" title="c31">fy</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15" title="TOPMETR:th.15">TOPMETR:15</a></i>;<br/>
<a NAME="E18:43_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c30">fx</font>,<font color="Maroon" title="c15">ft1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c16">ft2</font>,<font color="Maroon" title="c31">fy</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c18">r'</font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E14:43_1_1_1"><i><font color="Green" title="E40">A18</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E19:43_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c27">x</font>,<font color="Maroon" title="c28">y</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c18">r'</font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E17:43_1_1_1"><i><font color="Green" title="E41">A19</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:43_1_1_1"/><b>then </b><i><font color="Green" title="E42">A20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c27">x</font>,<font color="Maroon" title="c28">y</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">r'</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T23" title="XREAL_1:th.23">XREAL_1:23</a></i>;<br/>
<a NAME="E21:43_1_1_1"/>
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c27">x</font>,<font color="Maroon" title="c28">y</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">y</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E3:43_1_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="nagata_1.html#E8:43_1_1_1"><i><font color="Green" title="E38">A16</font></i></a>, <a class="txt" href="nagata_1.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<a NAME="E22:43_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c30">fx</font>,<font color="Maroon" title="c15">ft1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c27">x</font>,<font color="Maroon" title="c28">y</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">y</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15" title="TOPMETR:th.15">TOPMETR:15</a></i>;<br/>
<a NAME="E23:43_1_1_1"/><b>then </b><i><font color="Green" title="E43">A21</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</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"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c27">x</font>,<font color="Maroon" title="c28">y</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c30">fx</font>,<font color="Maroon" title="c15">ft1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c31">fy</font>,<font color="Maroon" title="c16">ft2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15" title="TOPMETR:th.15">TOPMETR:15</a></i>;<br/>
<a NAME="E24:43_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c27">x</font>,<font color="Maroon" title="c28">y</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c30">fx</font>,<font color="Maroon" title="c15">ft1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c31">fy</font>,<font color="Maroon" title="c16">ft2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c27">x</font>,<font color="Maroon" title="c28">y</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c18">r'</font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E14:43_1_1_1"><i><font color="Green" title="E40">A18</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E25:43_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c27">x</font>,<font color="Maroon" title="c28">y</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">r'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E23:43_1_1_1"><i><font color="Green" title="E43">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E26:43_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c18">r'</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c27">x</font>,<font color="Maroon" title="c28">y</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="real_2.html#T110" title="REAL_2:th.110">REAL_2:110</a></i>;<br/>
<a NAME="E27:43_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c27">x</font>,<font color="Maroon" title="c28">y</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c18">r'</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="real_2.html#T170" title="REAL_2:th.170">REAL_2:170</a></i>;<br/>
<a NAME="E28:43_1_1_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"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c27">x</font>,<font color="Maroon" title="c28">y</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tt</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">r'</font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E20:43_1_1_1"><i><font color="Green" title="E42">A20</font></i></a>, <a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></i>;<br/>
<a NAME="E29:43_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c14">Ftt</font>,<font color="Maroon" title="c29">Fxy1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">r'</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15" title="TOPMETR:th.15">TOPMETR:15</a></i>;<br/>
<a NAME="E30:43_1_1_1"/><b>then </b>
<font color="Maroon" title="c29">Fxy1</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="c14">Ftt</font>,<font color="Maroon" title="c17">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/>
<b>hence </b><a NAME="E31:43_1_1_1"/>
<font color="Maroon" title="c23">Fxy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">R</font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E7:43_1_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="nagata_1.html#E3:43_1_1_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="nagata_1.html#E5:43_1_1_1"><i><font color="Green" title="E37">A15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:43_1_1"/>
 ex <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">tt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">H</font> &amp; <font color="Maroon" title="c5">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="c13">R</font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E18:43_1_1"><i><font color="Green" title="E35">A13</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:43_1"/>
<font color="Maroon" title="c5">F'</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c8">tt</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T48" title="TMAP_1:th.48">TMAP_1:48</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:43"/><b>then </b>
<font color="Maroon" title="c5">F'</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T55" title="TMAP_1:th.55">TMAP_1:55</a></i>;<br/>
<b>hence </b><a NAME="E8:43"/>
<font color="Maroon" title="c3">F</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a>
 <b>by </b><i><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/>


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