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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">X0</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>



<b>thus </b><a NAME="E1:122"/>
( <font color="Maroon">X0</font> is  <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> implies  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="tmap_1.html#K8" target="_self">modified_with_respect_to</a> <font color="Maroon">X0</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:122_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">X0</font> is  <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<a NAME="E3:122_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tsep_1.html#D1">TSEP_1:def 1</a></i>;<br/>
<a NAME="E4:122_1"/><b>then </b>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T14" target="_self">Th14</a></i>;<br/>
<b>hence </b><a NAME="E5:122_1"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="tmap_1.html#K8" target="_self">modified_with_respect_to</a> <font color="Maroon">X0</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:122"/>
(  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="tmap_1.html#K8" target="_self">modified_with_respect_to</a> <font color="Maroon">X0</font> implies <font color="Maroon">X0</font> is  <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:122_2"/><i><font color="Green">E29</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="tmap_1.html#K8" target="_self">modified_with_respect_to</a> <font color="Maroon">X0</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:122_2_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 ;<br/><a NAME="E2:122_2_1"/><b>then </b>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:122_2_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tmap_1.html#T14" target="_self">Th14</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:122_2"/>
<font color="Maroon">X0</font> is  <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#D1">TSEP_1:def 1</a></i>;<br/>


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


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