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<b>let </b><font color="Maroon">X</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">Y</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">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">X</font> is <a href="compts_1.html#V2">compact</a>
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">Y</font> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 <b>and </b><br/><a NAME="E3:54"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>and </b><br/><a NAME="E4:54"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E5:54"/><i><font color="Green">E38</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E6:54"/><i><font color="Green">E39</font></i>: 
 <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/>
<b>then reconsider </b><font color="Maroon">f2</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <b>by </b><i>, <a class="ref" href="jgraph_1.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<a NAME="E8:54"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E9:54"/><i><font color="Green">E41</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E10:54"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T3">TOPMETR:3</a></i>;<br/>
<a NAME="E11:54"/>
<font color="Maroon">f2</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:54"/><b>then </b>
<font color="Maroon">f2</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i>, , <a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, <a class="ref" href="jgraph_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="compts_1.html#T26">COMPTS_1:26</a></i>;<br/>
<b>hence </b><a NAME="E13:54"/>
 ex <font color="Olive">f1</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> st <br/>( <font color="Maroon">f</font> <a href="funct_2.html#R1">=</a> <font color="Olive">f1</font> &amp; <font color="Olive">f1</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  )
 ;<br/>


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