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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:45"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="pscomp_1.html#V9">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>assume </b><a NAME="E2:45"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E5:45"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E1:45"><i><font color="Green">E23</font></i></a>, <a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/><a NAME="E1:45_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/><a NAME="E2:45_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:45_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="nagata_1.html#E4:45_1"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E6:45_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="urysohn3.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>

<a NAME="E1:45_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E4:45_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:45_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="urysohn3.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> )
 ;<br/>

<a NAME="E3:45_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="urysohn3.html#K8">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="urysohn3.html#K8">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E2:45"><i><font color="Green">E24</font></i></a>, <a class="txt" href="nagata_1.html#E1:45_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E7:45_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E2:45_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E9:45_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nagata_1.html#E7:45_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span>, <font color="Maroon">c<sub>21</sub></font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E11:45_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>20</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E9:45_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="nagata_1.html#E33"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="topmetr.html#T21">TOPMETR:21</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E12:45_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E5:45"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tmap_1.html#T55">TMAP_1:55</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E14:45_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E11:45_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tmap_1.html#T48">TMAP_1:48</a></i>;<br/>
<a NAME="E15:45_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E5:45"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tmap_1.html#T55">TMAP_1:55</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E17:45_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E11:45_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tmap_1.html#T48">TMAP_1:48</a></i>;<br/>
<a NAME="E18:45_1_1"/><i><font color="Green">E35</font></i>: 
( <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E4:45_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="nagata_1.html#E14:45_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="nagata_1.html#E17:45_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E19:45_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:45_1_1_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>25</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:45_1_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>26</sub></font>, <font color="Maroon">c<sub>27</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:45_1_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E3:45_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>26</sub></font>, <font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>27</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="nagata_1.html#E5:45_1_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="borsuk_1.html#K7">]</a></span> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E8:45_1_1_1"/><i><font color="Green">E38</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E2:45"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E9:45_1_1_1"/>
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
;<br/>
<a NAME="E10:45_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:45_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>23</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E5:45_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>29</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E13:45_1_1_1"/>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K6">/</a> 2 &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E14:45_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="nagata_1.html#E17:45_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="nagata_1.html#E11:45_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E14:45_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E15:45_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E3:45_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="nagata_1.html#E8:45_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="nagata_1.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E16:45_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15">TOPMETR:15</a></i>;<br/>
<a NAME="E17:45_1_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15">TOPMETR:15</a></i>;<br/>
<a NAME="E18:45_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E14:45_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E19:45_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E17:45_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:45_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T23">XREAL_1:23</a></i>;<br/>
<a NAME="E21:45_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E3:45_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="nagata_1.html#E8:45_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="nagata_1.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E22:45_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15">TOPMETR:15</a></i>;<br/>
<a NAME="E23:45_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15">TOPMETR:15</a></i>;<br/>
<a NAME="E24:45_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E14:45_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E25:45_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E23:45_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E26:45_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="real_2.html#T110">REAL_2:110</a></i>;<br/>
<a NAME="E27:45_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T170">REAL_2:170</a></i>;<br/>
<a NAME="E28:45_1_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E20:45_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="seq_2.html#T9">SEQ_2:9</a></i>;<br/>
<a NAME="E29:45_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>30</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15">TOPMETR:15</a></i>;<br/>
<a NAME="E30:45_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E31:45_1_1_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E7:45_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="nagata_1.html#E3:45_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="nagata_1.html#E5:45_1_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:45_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E18:45_1_1"><i><font color="Green">E35</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:45_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T48">TMAP_1:48</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:45"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T55">TMAP_1:55</a></i>;<br/>
<b>hence </b><a NAME="E8:45"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pscomp_1.html#V9">continuous</a>
 <b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>


</div>
