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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="E1:21"/><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="E2:21"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">D0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font>
 ;<br/>

<b>assume </b><a NAME="E3:21"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">D0</font> is <a href="tsp_1.html#V1" target="_self">T_0</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y1</font>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y1</font>;<br/><b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> )
 ;<br/><b>assume </b><a NAME="E2:21_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><b>reconsider </b><font color="Maroon">x0</font> = <font color="Maroon">x</font>, <font color="Maroon">y0</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y0</font> <b>by </b><i/>;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_1"/>
(  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> st <br/>( <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) or  ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> st <br/>( <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> ) )
 </span><b>by </b><i>, , <a class="txt" href="tsp_1.html#E1:21"><i><font color="Green">E17</font></i></a>, <a class="txt" href="tsp_1.html#E2:21"><i><font color="Green">E18</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_1_1"/>
 ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> st <br/>( <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font><b> such that </b><br/><a NAME="E3:21_1_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:21_1_1_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 ;<br/><b>reconsider </b><font color="Maroon">V1</font> = <font color="Maroon">V</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="21_1_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">V1</font> = <font color="Maroon">V1</font>;<br/><a NAME="E1:21_1_1_1_1_1"/>
<font color="Maroon">V1</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="txt" href="tsp_1.html#E3:21"><i><font color="Green">E19</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>hence </b><a NAME="E2:21_1_1_1_1_1"/>
<font color="Maroon">V1</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="E3:21_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> )
 <b>by </b><i><a class="txt" href="tsp_1.html#E1:21_1"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:21_1_1_1_1"/>
(  ex <font color="Olive">V1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y1</font> st <br/>( <font color="Olive">V1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V1</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">V1</font> ) or  ex <font color="Olive">W1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y1</font> st <br/>( <font color="Olive">W1</font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W1</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_1_2"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> st <br/>( <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font><b> such that </b><br/><a NAME="E3:21_1_1_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:21_1_1_1_2"/><i><font color="Green">E26</font></i>: 
( not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 ;<br/><b>reconsider </b><font color="Maroon">W1</font> = <font color="Maroon">W</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="21_1_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">W1</font> = <font color="Maroon">W1</font>;<br/><a NAME="E1:21_1_1_1_2_1"/>
<font color="Maroon">W1</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="txt" href="tsp_1.html#E3:21"><i><font color="Green">E19</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>hence </b><a NAME="E2:21_1_1_1_2_1"/>
<font color="Maroon">W1</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="E3:21_1_1_1_2_1"/>
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> )
 <b>by </b><i><a class="txt" href="tsp_1.html#E1:21_1"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:21_1_1_1_2"/>
(  ex <font color="Olive">V1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y1</font> st <br/>( <font color="Olive">V1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V1</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">V1</font> ) or  ex <font color="Olive">W1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y1</font> st <br/>( <font color="Olive">W1</font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W1</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:21"/>
<font color="Maroon">D1</font> is <a href="tsp_1.html#V1" target="_self">T_0</a>
 <b>by </b><i/>;<br/>


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