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





<b>assume </b><b>that </b><br/><a NAME="E1:63"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>and </b><br/><a NAME="E2:63"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">g</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 ;<br/>

<a NAME="E3:63"/><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> &amp; <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <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/>
<a NAME="E4:63"/><i><font color="Green">E41</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">V</font> &amp; <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a> )
 
<b>by </b><i><a class="txt" href="tops_2.html#E1:63"><i><font color="Green">E33</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E5:63"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="txt" href="tops_2.html#E2:63"><i><font color="Green">E34</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D5" target="_self">TOPS_2:def 5</a><br/>

<b>thus </b><a NAME="E6:63"/>
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="tops_2.html#E2:63"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tops_2.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_1.html#T46">FUNCT_1:46</a></i>;<br/>

<b>thus </b><a NAME="E7:63"/>
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="tops_2.html#E2:63"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tops_2.html#T11" target="_self">Th11</a>, </i>;<br/>

<a NAME="E8:63"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="tops_2.html#E2:63"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tops_2.html#T11" target="_self">Th11</a>, </i>;<br/>
<b>hence </b><a NAME="E9:63"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="tops_2.html#E2:63"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tops_2.html#T11" target="_self">Th11</a>, </i>;<br/>


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