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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">T</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">S0</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">T0</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>







<b>assume </b><a NAME="E1:12"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 ;<br/>

<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S0</font>,<font color="Maroon">T0</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:12"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f</font> <a href="tmap_1.html#K2">|</a> <font color="Maroon">S0</font>
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">g</font> is <a href="funct_2.html#V2">onto</a>
 ;<br/>

<b>thus </b><a NAME="E4:12"/>
 <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">S0</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a><br/>

<b>thus </b><a NAME="E5:12"/><i><font color="Green">E50</font></i>: 
 <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">T0</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>

<a NAME="E6:12"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E7:12"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S0</font>
 
<b>by </b><i>, <a class="ref" href="tmap_1.html#D3">TMAP_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:12"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/>

<a NAME="E9:12"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E10:12"/><b>then </b>
<font color="Maroon">g</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S0</font>,<font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="tmap_1.html#T68">TMAP_1:68</a></i>;<br/>
<b>hence </b><a NAME="E11:12"/>
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i/>;<br/>

<a NAME="E12:12"/><i><font color="Green">E54</font></i>: 
 <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">T</font>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<i><font color="Green">E55</font></i>: <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S0</font> = 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
<b>by </b><i>, <a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T0</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>
;<br/>
<i><font color="Green">E69</font></i>: <font color="Maroon">g</font> <a href="tops_2.html#NK3" target="_self">"</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S0</font></span>)</span> <a href="funct_1.html#K2">"</a> 
<b>by </b><i>, , , <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
<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#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S0</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T40">RFUNCT_2:40</a></i>
<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="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T0</font>
<b>by </b><i>, , , <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<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="tmap_1.html#K2">|</a> <font color="Maroon">T0</font>
<b>by </b><i><a class="ref" href="tmap_1.html#D3">TMAP_1:def 3</a></i>
;<br/>
<a NAME="E15:12"/>
<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>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E16:12"/><b>then </b>
<font color="Maroon">g</font> <a href="tops_2.html#NK3" target="_self">"</a>  is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T0</font>,<font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="tmap_1.html#T68">TMAP_1:68</a></i>;<br/>
<b>hence </b><a NAME="E17:12"/>
<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/>;<br/>


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