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<b>let </b><font color="Maroon" title="c3">S</font>, <font color="Maroon" title="c4">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="c5">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font>;<br/>



<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E11">A1</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>
 ;<br/>

<a NAME="E2:28"/><i><font color="Green" title="E12">A2</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<b>assume </b><a NAME="E3:28"/>
<font color="Maroon" title="c5">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 ;<br/>

<a NAME="E4:28"/><b>then </b>
 <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="c4">T</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E5:28"/><b>then </b><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> 
 
<b>by </b><i><a class="txt" href="topreala.html#E1:28"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/>
<b>thus </b><a NAME="E6:28"/>
( <font color="Maroon" title="c5">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> implies <font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c5">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D2" title="T_0TOPSP:def.2">T_0TOPSP:def 2</a><br/>

<b>assume </b><a NAME="E2:28_1"/>
<font color="Maroon" title="c6">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E3:28_1"/><b>then </b>
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="topreala.html#E2:28"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="topreala.html#E1:28_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>
<b>hence </b><a NAME="E4:28_1"/>
<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="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="topreala.html#E1:28"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="topreala.html#E5:28"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="funct_1.html#T155" title="FUNCT_1:th.155">FUNCT_1:155</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E7:28"/><i><font color="Green" title="E14">A5</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T</font>;<br/><b>assume </b><a NAME="E1:28_2"/>
<font color="Maroon" title="c6">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/><a NAME="E2:28_2"/><b>then </b>
<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="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="topreala.html#E7:28"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="t_0topsp.html#D2" title="T_0TOPSP:def.2">T_0TOPSP:def 2</a></i>;<br/><b>hence </b><a NAME="E3:28_2"/>
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="topreala.html#E1:28"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="topreala.html#E5:28"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="funct_1.html#T155" title="FUNCT_1:th.155">FUNCT_1:155</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:28"/>
<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="topreala.html#E2:28"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>


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