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<b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">Y0</font> be  non <a href="tex_2.html#V2" target="_self">proper</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y</font>;<br/>



<b>consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E2:34"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font>
 <b>and </b><br/><a NAME="E3:34"/><i><font color="Green">E22</font></i>: 
not <font color="Maroon">A</font> is <a href="tex_2.html#V1" target="_self">proper</a>
 <b>by </b><i/>;<br/>
<a NAME="E4:34"/><i><font color="Green">E24</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:34"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">D</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:34_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font>
 ;<br/><b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">D</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> ;<br/><b>consider </b><font color="Maroon">E</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:34_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">E</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E5:34_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y0</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/><b>thus </b><a NAME="E6:34_1_1"/>
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="tex_2.html#D1" target="_self">Def1</a>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:34_1"/><b>then </b><i><font color="Green">E50</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">D</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:34_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 ;<br/><b>then 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">Y</font> ;<br/><b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">E</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> <b>by </b><i>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">E</font> = <font color="Maroon">E</font>;<br/><b>thus </b><a NAME="E1:34_1_2_1"/>
( <font color="Maroon">E</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y0</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tex_2.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:34_1_2"/>
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:34_1"/><b>then </b>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:34_1"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:34"/>
 <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">Y</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> #)
 <b>by </b><i>, , </i>;<br/>


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