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<b>let </b><font color="Maroon">X</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">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="tdlat_3.html#V2">anti-discrete</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/><b>assume </b><a NAME="E1:120_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_1_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> )
 </span><b>by </b><i>, <a class="ref" href="tdlat_3.html#T21">TDLAT_3:21</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1_1_1"><b>suppose </b></a><a NAME="E1:120_1_1_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:120_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T171">RELAT_1:171</a></i>;<br/><b>hence </b><a NAME="E3:120_1_1_1_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1_1_2"><b>suppose </b></a><a NAME="E1:120_1_1_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 ;<br/><div class="add"><a NAME="E2:120_1_1_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E3:120_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tops_2.html#T52">TOPS_2:52</a></i>;<br/><b>hence </b><a NAME="E4:120_1_1_1_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:120_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:120"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>


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