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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</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>5</sub></font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>







<b>assume </b><a NAME="E1:34"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:34"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K10">Base-Appr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T54">BORSUK_1:54</a></i>;<br/>
<a NAME="E3:34"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> <a href="weierstr.html#K1">"</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K10">Base-Appr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_2.html#T9" target="_self">Th9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> <a href="weierstr.html#K1">"</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K10">Base-Appr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> as   <a href="struct_0.html#NM3">Subset-Family</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>2</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<a NAME="E5:34"/>
for <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>2</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <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>2</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E3:34_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_1.html#K10">Base-Appr</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="weierstr.html#D1">WEIERSTR:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> 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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<b>thus </b><a NAME="E5:34_1"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="borsuk_2.html#E3:34_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="borsuk_2.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:34"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:34"/>
<span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="borsuk_2.html#E3:34"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tops_2.html#T26">TOPS_2:26</a></i>;<br/>


</div>
