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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="E1:64_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 ;<br/><a NAME="E2:64_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:64_1"/><i><font color="Green">E40</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:64_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, </i>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:64_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">P</font>
<b>by </b><i><a class="txt" href="tops_2.html#E2:64_1"><i><font color="Green">E34</font></i></a>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font>
<b>by </b><i><a class="txt" href="tops_2.html#E2:64_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T154">FUNCT_1:154</a></i>
;<br/><b>hence </b><a NAME="E3:64_1_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T11" target="_self">Th11</a>, <a class="ref" href="tops_2.html#T12" target="_self">Th12</a>, <a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E6:64_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><a NAME="E7:64_1"/><b>then </b><i><font color="Green">E43</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">P</font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="tops_2.html#E1:64_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/><a NAME="E8:64_1"/><i><font color="Green">E45</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">P</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="tops_2.html#E2:64_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T152">FUNCT_1:152</a></i>;<br/><a NAME="E9:64_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E10:64_1"/><b>then </b>
<font color="Maroon">P</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">P</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/><b>hence </b><a NAME="E11:64_1"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T12" target="_self">Th12</a>, <a class="txt" href="tops_2.html#E13"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:64"/><i><font color="Green">E46</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> )
 ;<br/>

<b>assume </b><a NAME="E3:64"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> holds <br/> ( <font color="Olive">P</font> is <a href="pre_topc.html#V4">closed</a> iff <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">P</font> is <a href="pre_topc.html#V4">closed</a> )
 ;<br/>

<a NAME="E4:64"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</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="E1:64_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>set </b><font color="Maroon">D</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font>;<br/>
<a NAME="E2:64_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>;<br/>
<b>hence </b><a NAME="E3:64_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:64"/>
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</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="E1:64_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">B</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T154">FUNCT_1:154</a></i>
;<br/>
<b>hence </b><a NAME="E3:64_3"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:64"/>
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, , </i>;<br/>


</div>
