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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/>

<b>assume </b><a NAME="E1:36"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel14.html#D1" target="_self">WAYBEL14:def 1</a><br/>

<b>assume </b><a NAME="E2:36"/><i><font color="Green">E27</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p1">(<span class="default"><a href="waybel11.html#K2">Scott-Convergence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E3:36"/><i><font color="Green">E28</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p1">(<span class="default"><a href="waybel11.html#K2">Scott-Convergence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#T32">WAYBEL11:32</a></i>;<br/>
<a NAME="E4:36"/><i><font color="Green">E29</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:36"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow_6.html#D27">YELLOW_6:def 27</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="waybel_2.html#K5">sup_op</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E5:36"/><i><font color="Green">E30</font></i>: 
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>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</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>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</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/>
<a NAME="E6:36"/><b>then </b><i><font color="Green">E31</font></i>: 
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>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="txt" href="waybel14.html#E4:36"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="waybel_2.html#K5">sup_op</a> <font color="Maroon">c<sub>1</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>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel14.html#E4:36"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E8:36"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R1">=</a>  <a href="waybel_2.html#K5">sup_op</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E9:36"/>
for <font color="Olive">b<sub>1</sub></font> being  <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>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> holds <font color="Maroon">c<sub>4</sub></font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:36_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="waybel14.html#E5:36"><i><font color="Green">E30</font></i></a>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tmap_1.html#D2">TMAP_1:def 2</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E4:36_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E5:36_1"/><i><font color="Green">E34</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T51">TOPS_1:51</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel14.html#E2:36"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow_6.html#D27">YELLOW_6:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>10</sub></font> </span>}</span>  as    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="txt" href="waybel14.html#E1:36"><i><font color="Green">E26</font></i></a>, <a class="ref" href="waybel11.html#T44">WAYBEL11:44</a></i>;<br/>
<a NAME="E8:36_1"/>
<font color="Maroon">c<sub>11</sub></font> is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:36"><i><font color="Green">E27</font></i></a>, <a class="txt" href="waybel14.html#E3:36"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel14.html#T21" target="_self">Th21</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E10:36_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E4:36_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel14.html#E5:36_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E12:36_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E10:36_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel14.html#E2:36"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow_6.html#D27">YELLOW_6:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a> <b>by </b><i><a class="txt" href="waybel14.html#E1:36"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E15:36_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E16:36_1"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a></i>;<br/>
<i><font color="Green">E38</font></i>: <font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="waybel14.html#E2:36_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel_2.html#D5">WAYBEL_2:def 5</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="yellow_4.html#K2">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel14.html#E16:36_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="waybel_2.html#T4">WAYBEL_2:4</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> =  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>14</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> =  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>15</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="yellow_4.html#K2">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E19:36_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="yellow_4.html#K2">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E1:36"><i><font color="Green">E26</font></i></a>, <a class="txt" href="waybel14.html#E12:36_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel14.html#E17:36_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="waybel_4.html#T54">WAYBEL_4:54</a></i>;<br/>
<a NAME="E20:36_1"/>
<span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="yellow_4.html#K2">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>15</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#D3">YELLOW_4:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E22:36_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E19:36_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E23:36_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>15</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E22:36_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/>
<a NAME="E24:36_1"/><i><font color="Green">E42</font></i>: 
(  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>22</sub></font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E1:36"><i><font color="Green">E26</font></i></a>, <a class="ref" href="waybel11.html#T36">WAYBEL11:36</a></i>;<br/>
<a NAME="E25:36_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>22</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E23:36_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> =  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>24</sub></font> =  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>22</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel14.html#E2:36"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow_6.html#D27">YELLOW_6:def 27</a></i>;<br/>
<a NAME="E27:36_1"/>
(  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E24:36_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E28:36_1"/><b>then </b><i><font color="Green">E44</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>24</sub></font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:36"><i><font color="Green">E27</font></i></a>, <a class="txt" href="waybel14.html#E3:36"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1"><a href="yellow_3.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span><a href="yellow_3.html#K12">:]</a></span> as   <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>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="txt" href="waybel14.html#E6:36"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E30:36_1"/>
<font color="Maroon">c<sub>25</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E28:36_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>
<a NAME="E31:36_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<a NAME="E32:36_1"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E25:36_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>25</sub></font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="waybel14.html#E2:36_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel14.html#E31:36_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>26</sub></font>
;<br/>

<a NAME="E34:36_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="yellow_4.html#K2">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_2.html#T35">WAYBEL_2:35</a></i>;<br/>
<a NAME="E35:36_1"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="yellow_4.html#K2">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel14.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E36:36_1"/>
 <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E19:36_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel14.html#E22:36_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel14.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E37:36_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>26</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E12:36_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel14.html#E34:36_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel14.html#E35:36_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E38:36_1"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E39:36_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E10:36_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E40:36_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>26</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E37:36_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:36"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49">TMAP_1:49</a></i>;<br/>


</div>
