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<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c3">V</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c5">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">V</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:63"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>and </b><br/><a NAME="E2:63"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 ;<br/>

<a NAME="E3:63"/><i><font color="Green" title="E34">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c4">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c4">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c4">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 
<b>by </b><i><a class="txt" href="tops_2.html#E1:63"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" target="_self" title="TOPS_2:def.5">Def5</a></i>;<br/>
<a NAME="E4:63"/><i><font color="Green" title="E35">A4</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c5">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c5">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c5">g</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 
<b>by </b><i><a class="txt" href="tops_2.html#E2:63"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="tops_2.html#D5" target="_self" title="TOPS_2:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E5:63"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">V</font> )
 <b>by </b><i><a class="txt" href="tops_2.html#E3:63"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a>, <a class="ref" href="relat_1.html#T47" title="RELAT_1:th.47">RELAT_1:47</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D5" target="_self" title="TOPS_2:def.5">TOPS_2:def 5</a><br/>

<b>thus </b><a NAME="E6:63"/>
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="tops_2.html#E3:63"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="tops_2.html#E4:63"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#T46" title="FUNCT_1:th.46">FUNCT_1:46</a></i>;<br/>

<b>thus </b><a NAME="E7:63"/>
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="tops_2.html#E3:63"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="tops_2.html#E4:63"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="tops_2.html#T58" target="_self" title="TOPS_2:th.58">Th58</a></i>;<br/>

<a NAME="E8:63"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="tops_2.html#E3:63"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="tops_2.html#E4:63"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="tops_2.html#T58" target="_self" title="TOPS_2:th.58">Th58</a></i>;<br/>
<b>hence </b><a NAME="E9:63"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="tops_2.html#E3:63"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="tops_2.html#E4:63"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="tops_2.html#T66" target="_self" title="TOPS_2:th.66">Th66</a></i>;<br/>


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