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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">X1</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y</font>, <font color="Maroon">X2</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y</font>;<br/>



<b>set </b><font color="Maroon">A1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>;<br/>
<b>set </b><font color="Maroon">A2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>;<br/>
<a NAME="E1:104"/><i><font color="Green">E13</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X1</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X2</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font> )
 
;<br/>
<b>assume </b><a NAME="E2:104"/><i><font color="Green">E20</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 ;<br/>

<a NAME="E3:104"/>
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X1</font> holds <br/> ( <font color="Olive">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X1</font> iff  ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X2</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X2</font> &amp; <font color="Olive">P</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X1</font>;<br/>

<b>thus </b><a NAME="E1:104_1"/>
( <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X1</font> implies  ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X2</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X2</font> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:104_1_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">R</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:104_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">R</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="E4:104_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T17" target="_self">Th17</a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">R</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X2</font> <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>take </b>
<font color="Maroon">Q</font>
;<br/>

<b>thus </b><a NAME="E6:104_1_1"/>
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T17" target="_self">Th17</a>, <a class="ref" href="tex_4.html#T19" target="_self">Th19</a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>

<font color="Maroon">Q</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> = 
<font color="Maroon">R</font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">R</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
<b>by </b><i><a class="ref" href="tex_4.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E8:104_1_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T20" target="_self">Th20</a></i>;<br/>


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

<b>given </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X2</font><b> such that </b><a NAME="E2:104_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X2</font>
 <b>and </b><br/><a NAME="E3:104_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 ;<br/>

<b>consider </b><font color="Maroon">R</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E5:104_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">R</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="E6:104_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X2</font></span>)</span>
 <b>by </b><i><a class="ref" href="tex_4.html#T19" target="_self">Th19</a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<font color="Maroon">P</font> = 
<font color="Maroon">R</font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font></span>)</span>
<b>by </b><i><a class="ref" href="tex_4.html#T20" target="_self">Th20</a>, <a class="ref" href="tex_4.html#T22" target="_self">Th22</a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">R</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
<b>by </b><i><a class="ref" href="tex_4.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E8:104_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X1</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T17" target="_self">Th17</a>, <a class="ref" href="tex_4.html#T21" target="_self">Th21</a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:104"/>
<font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T17" target="_self">Th17</a>, <a class="ref" href="tex_4.html#T18" target="_self">Th18</a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>


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