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<div class="add">

<b>thus </b><a NAME="E1:33_1_1"/>
( <font color="Maroon">F</font> is <a href="pre_topc.html#V5">continuous</a> implies for <font color="Olive">W</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">G</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">W</font>  ex <font color="Olive">H</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">W</font> st <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">F</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon">W</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font>;<br/>



<a NAME="E2:33_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E3:33_1_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<a NAME="E4:33_1_1_1"/>
<font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">G</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T12" target="_self">Th12</a>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<a NAME="E5:33_1_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">H</font> = <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">G</font></span>)</span> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">W</font> <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<a NAME="E7:33_1_1_1"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E8:33_1_1_1"/><b>then </b>
<font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/>
<b>hence </b><a NAME="E9:33_1_1_1"/>
<font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:33_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">W</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">G</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">W</font>  ex <font color="Olive">H</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">W</font> st <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/><b>assume </b><a NAME="E1:33_1_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">P1</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:33_1_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font> implies  ex <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">P</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_1_2_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">W</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E3:33_1_1_2_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<a NAME="E4:33_1_1_2_1_1"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<a NAME="E5:33_1_1_2_1_1"/><b>then </b>
<font color="Maroon">P1</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">H</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E7:33_1_1_2_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T12" target="_self">Th12</a></i>;<br/>
<b>take </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">H</font>
;<br/>

<b>thus </b><a NAME="E8:33_1_1_2_1_1"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E9:33_1_1_2_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E2:33_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/>
<a NAME="E10:33_1_1_2_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:33_1_1_2_1_1"/><b>then </b>
<font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/>
<a NAME="E12:33_1_1_2_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E1:33_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:33_1_1_2_1_1"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<b>hence </b><a NAME="E14:33_1_1_2_1_1"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E1:33_1_1_2_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<b>thus </b><a NAME="E15:33_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:33_1_1_2_1"/>
(  ex <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">P</font> ) implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_2"/>
<font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T57">TOPS_1:57</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:33_1_1"/>
<font color="Maroon">F</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div>
