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<b>let </b><font color="Maroon">T1</font> be   <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">T2</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green">E37</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">A1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font><br/> for <font color="Olive">A2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font>  st <font color="Olive">A1</font> <a href="hidden.html#R1">=</a> <font color="Olive">A2</font> holds <br/> <a href="pre_topc.html#K4">Cl</a> <font color="Olive">A1</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">A2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:14_1"/>
( <font color="Maroon">A</font> is  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font> implies <font color="Maroon">A</font> is  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_1"/>
<font color="Maroon">A</font> is  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A</font> as  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font> ;<br/>
<b>reconsider </b><font color="Maroon">A2</font> = <font color="Maroon">A1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> <b>by </b><i/>;<br/>
<a NAME="E4:14_1_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E5:14_1_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:14_1_1"/>
<font color="Maroon">A</font> is  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font>
 ;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:14_1"/>
<font color="Maroon">A</font> is  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font>
 ;<br/><b>then reconsider </b><font color="Maroon">A2</font> = <font color="Maroon">A</font> as  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> ;<br/><b>reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font> <b>by </b><i/>;<br/><a NAME="E5:14_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/><a NAME="E6:14_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:14_1"/>
<font color="Maroon">A</font> is  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E4:14"/><b>then </b>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</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">T2</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> #)
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:14"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font>
 ;<br/>


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