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<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">S0</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c4">T0</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">T</font>;<br/><b>let </b><font color="Maroon" title="c5">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>;<br/>







<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c6">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">S0</font>,<font color="Maroon" title="c4">T0</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:12"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c6">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c3">S0</font>
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c6">g</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 ;<br/>

<b>thus </b><a NAME="E4:12"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">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">S0</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a><br/>

<b>thus </b><a NAME="E5:12"/><i><font color="Green" title="E10">A4</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">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="c4">T0</font>
 <b>by </b><i><a class="txt" href="jordan16.html#E3:12"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>

<a NAME="E6:12"/><i><font color="Green" title="E11">A5</font></i>: 
<font color="Maroon" title="c5">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="jordan16.html#E1:12"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E7:12"/><i><font color="Green" title="E12">A6</font></i>: 
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S0</font>
 
<b>by </b><i><a class="txt" href="jordan16.html#E2:12"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="tmap_1.html#D3" title="TMAP_1:def.3">TMAP_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:12"/><i><font color="Green" title="E13">A7</font></i>: 
<font color="Maroon" title="c6">g</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="jordan16.html#E6:12"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></i>;<br/>

<a NAME="E9:12"/>
<font color="Maroon" title="c5">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jordan16.html#E1:12"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E10:12"/><b>then </b>
<font color="Maroon" title="c6">g</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">S0</font>,<font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="txt" href="jordan16.html#E2:12"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="tmap_1.html#T68" title="TMAP_1:th.68">TMAP_1:68</a></i>;<br/>
<b>hence </b><a NAME="E11:12"/>
<font color="Maroon" title="c6">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="jordan16.html#T12" target="_self" title="JORDAN16:th.12">Th12</a></i>;<br/>

<a NAME="E12:12"/><i><font color="Green" title="E14">A8</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">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">T</font>
 
<b>by </b><i><a class="txt" href="jordan16.html#E1:12"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<i><font color="Green" title="E15">A9</font></i>: <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S0</font> = 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">g</font>
<b>by </b><i><a class="txt" href="jordan16.html#E7:12"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T0</font>
<b>by </b><i><a class="txt" href="jordan16.html#E3:12"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>
;<br/>
<i><font color="Green" title="E16">A10</font></i>: <font color="Maroon" title="c6">g</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S0</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
<b>by </b><i><a class="txt" href="jordan16.html#E5:12"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="jordan16.html#E7:12"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="jordan16.html#E8:12"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S0</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan16.html#E6:12"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="rfunct_2.html#T40" title="RFUNCT_2:th.40">RFUNCT_2:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T0</font>
<b>by </b><i><a class="txt" href="jordan16.html#E6:12"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="jordan16.html#E12:12"><i><font color="Green" title="E14">A8</font></i></a>, <a class="txt" href="jordan16.html#E13:12"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c4">T0</font>
<b>by </b><i><a class="ref" href="tmap_1.html#D3" title="TMAP_1:def.3">TMAP_1:def 3</a></i>
;<br/>
<a NAME="E15:12"/>
<font color="Maroon" title="c5">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="jordan16.html#E1:12"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E16:12"/><b>then </b>
<font color="Maroon" title="c6">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> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">T0</font>,<font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="txt" href="jordan16.html#E14:12"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="tmap_1.html#T68" title="TMAP_1:th.68">TMAP_1:68</a></i>;<br/>
<b>hence </b><a NAME="E17:12"/>
<font color="Maroon" title="c6">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="ref" href="jordan16.html#T12" target="_self" title="JORDAN16:th.12">Th12</a></i>;<br/>


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