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<b>let </b><font color="Maroon">X1</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">X2</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/>



<b>assume </b><a NAME="E1:76"/><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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 ;<br/>

<b>assume </b><a NAME="E2:76"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">C1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X1</font><br/> for <font color="Olive">C2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X2</font>  st <font color="Olive">C1</font> <a href="hidden.html#R1">=</a> <font color="Olive">C2</font> holds <br/>( <font color="Olive">C1</font> is <a href="pre_topc.html#V4">closed</a> iff <font color="Olive">C2</font> is <a href="pre_topc.html#V4">closed</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">C1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X1</font>;<br/><b>let </b><font color="Maroon">C2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X2</font>;<br/><b>assume </b><a NAME="E1:76_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font>
 ;<br/><b>thus </b><a NAME="E2:76_1"/>
( <font color="Maroon">C1</font> is <a href="pre_topc.html#V3">open</a> implies <font color="Maroon">C2</font> is <a href="pre_topc.html#V3">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_1_1"/>
<font color="Maroon">C1</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:76_1_1"/><b>then </b>
( <font color="Maroon">C1</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">C1</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i><a class="ref" href="tops_3.html#T39" target="_self">Th39</a>, <a class="ref" href="tops_3.html#T41" target="_self">Th41</a>, <a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>
<a NAME="E3:76_1_1"/><b>then </b>
<font color="Maroon">C2</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T40" target="_self">Th40</a></i>;<br/>
<b>hence </b><a NAME="E4:76_1_1"/>
<font color="Maroon">C2</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:76_1"/>
( <font color="Maroon">C2</font> is <a href="pre_topc.html#V3">open</a> implies <font color="Maroon">C1</font> is <a href="pre_topc.html#V3">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_1_2"/>
<font color="Maroon">C2</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:76_1_2"/><b>then </b>
( <font color="Maroon">C2</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">C1</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i><a class="ref" href="tops_3.html#T39" target="_self">Th39</a>, <a class="ref" href="tops_3.html#T41" target="_self">Th41</a>, <a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>
<a NAME="E3:76_1_2"/><b>then </b>
<font color="Maroon">C1</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T40" target="_self">Th40</a></i>;<br/>
<b>hence </b><a NAME="E4:76_1_2"/>
<font color="Maroon">C1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:76"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X1</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">X2</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X2</font> #)
 <b>by </b><i><a class="ref" href="tops_3.html#T39" target="_self">Th39</a>, <a class="ref" href="tops_3.html#T36" target="_self">Th36</a></i>;<br/>


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