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<b>let </b><font color="Maroon">Y0</font> be    <a href="pre_topc.html#L1">TopStruct</a> , <font color="Maroon">Y1</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">D0</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font>;<br/><b>let </b><font color="Maroon">D1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y1</font>;<br/>





<b>assume </b><a NAME="E1:65"/><i><font color="Green">E17</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y1</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y1</font> #)
 ;<br/>

<b>assume </b><a NAME="E2:65"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">D0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font>
 ;<br/>

<b>assume </b><a NAME="E3:65"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">D0</font> is <a href="tex_2.html#V3" target="_self">discrete</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y1</font>;<br/><b>reconsider </b><font color="Maroon">E</font> = <font color="Maroon">D</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> <b>by </b><i/>;<br/><b>assume </b><a NAME="E2:65_1"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D1</font>
 ;<br/><b>then consider </b><font color="Maroon">G0</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font><b> such that </b><br/><a NAME="E4:65_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">G0</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:65_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">D0</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 <b>by </b><i>, <a class="txt" href="tex_2.html#E2:65"><i><font color="Green">E22</font></i></a>, <a class="txt" href="tex_2.html#E1:65"><i><font color="Green">E17</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y1</font> <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">G</font> = <font color="Maroon">G</font>;<br/><a NAME="E1:65_1_1"/>
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y1</font>
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>hence </b><a NAME="E2:65_1_1"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>thus </b><a NAME="E3:65_1_1"/>
<font color="Maroon">D1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="txt" href="tex_2.html#E3:65"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:65_1"/>
 ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y1</font> st <br/>( <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">D1</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:65"/>
<font color="Maroon">D1</font> is <a href="tex_2.html#V3" target="_self">discrete</a>
 <b>by </b><i><a class="txt" href="tex_2.html#E1:65"><i><font color="Green">E17</font></i></a></i>;<br/>


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