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<b>let </b><font color="Maroon">T</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:45_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/><b>let </b><font color="Maroon">P1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E2:45_1"/>
<font color="Maroon">P1</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E3:45_1"/><b>then </b>
<font color="Maroon">P1</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/><a NAME="E4:45_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="subset_1.html#K3">`</a> </span>)</span> = 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">P1</font></span>)</span>
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>
<br/>.= 
<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="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T138">FUNCT_1:138</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>
;<br/><b>hence </b><a NAME="E6:45_1"/>
<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#T30">TOPS_1:30</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:45"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">P1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>  st <font color="Olive">P1</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Olive">P1</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>let </b><font color="Maroon">P1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a><br/>

<b>assume </b><a NAME="E3:45"/>
<font color="Maroon">P1</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E4:45"/><b>then </b>
<font color="Maroon">P1</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<a NAME="E5:45"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="subset_1.html#K3">`</a> </span>)</span> = 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">P1</font></span>)</span>
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>
<br/>.= 
<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="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T138">FUNCT_1:138</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E7:45"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>


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