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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>thus </b><a NAME="E1:11"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="decomp_1.html#K6" target="_self">^alpha</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K11" target="_self">D(c,alpha)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="decomp_1.html#K6" target="_self">^alpha</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K11" target="_self">D(c,alpha)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:11_1"/><b>then </b><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="decomp_1.html#K6" target="_self">^alpha</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K11" target="_self">D(c,alpha)</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:11_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is    <a href="decomp_1.html#M1" target="_self">alpha-set</a> of <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:11_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E6:11_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is    <a href="decomp_1.html#M1" target="_self">alpha-set</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>
<a NAME="E7:11_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K3" target="_self">alphaInt</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E6:11_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="decomp_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E8:11_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="tops_1.html#K1">Int</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K3" target="_self">alphaInt</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="decomp_1.html#E2:11_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:11_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E11:11_1"/><i><font color="Green">E18</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K3" target="_self">alphaInt</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>
<a NAME="E12:11_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E5:11_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="decomp_1.html#E7:11_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="decomp_1.html#E10:11_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="decomp_1.html#E11:11_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tops_1.html#T51">TOPS_1:51</a></i>;<br/>
<b>hence </b><a NAME="E13:11_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="decomp_1.html#E10:11_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:11"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:11"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E2:11"><i><font color="Green">E13</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:11"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<a NAME="E6:11"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E7:11"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E5:11"><i><font color="Green">E14</font></i></a>, <a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/>
<a NAME="E8:11"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is    <a href="decomp_1.html#M1" target="_self">alpha-set</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="decomp_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E9:11"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is    <a href="decomp_1.html#M1" target="_self">alpha-set</a> of <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
;<br/>
<a NAME="E10:11"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="decomp_1.html#K6" target="_self">^alpha</a> 
 
;<br/>
<a NAME="E11:11"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K3" target="_self">alphaInt</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E5:11"><i><font color="Green">E14</font></i></a>, <a class="txt" href="decomp_1.html#E8:11"><i><font color="Green">E15</font></i></a>, <a class="ref" href="decomp_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E12:11"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="tops_1.html#K1">Int</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="decomp_1.html#K3" target="_self">alphaInt</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
;<br/>
<a NAME="E13:11"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K11" target="_self">D(c,alpha)</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E14:11"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="decomp_1.html#K6" target="_self">^alpha</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K11" target="_self">D(c,alpha)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="decomp_1.html#E10:11"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div>
