<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>thus </b><a NAME="E1:13"/>
<span class="p1">(<span class="default"><a href="decomp_1.html#K8" title="DECOMP_1:func.8">PO</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K12" title="DECOMP_1:func.12">D(c,p)</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K8" title="DECOMP_1:func.8">PO</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K12" title="DECOMP_1:func.12">D(c,p)</a> <font color="Maroon" title="c1">T</font></span>)</span>
 ;<br/>

<a NAME="E2:13_1"/><b>then </b><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K8" title="DECOMP_1:func.8">PO</a> <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K12" title="DECOMP_1:func.12">D(c,p)</a> <font color="Maroon" title="c1">T</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:13_1"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E5:13_1"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c3">A</font> is <a href="decomp_1.html#V2" title="DECOMP_1:attr.2">pre-open</a>
 ;<br/>
<a NAME="E6:13_1"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K2" title="DECOMP_1:func.2">pInt</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E5:13_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="decomp_1.html#T4" target="_self" title="DECOMP_1:th.4">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">Z</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E8:13_1"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Z</font>
 <b>and </b><br/><a NAME="E9:13_1"/><i><font color="Green" title="E20">A6</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K2" title="DECOMP_1:func.2">pInt</a> <font color="Maroon" title="c4">Z</font>
 <b>by </b><i><a class="txt" href="decomp_1.html#E2:13_1"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/>
<a NAME="E10:13_1"/>
<font color="Maroon" title="c4">Z</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E4:13_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="decomp_1.html#E6:13_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="decomp_1.html#E8:13_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="decomp_1.html#E9:13_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="tops_1.html#T51" title="TOPS_1:th.51">TOPS_1:51</a></i>;<br/>
<b>hence </b><a NAME="E11:13_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="decomp_1.html#E8:13_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:13"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">K</font> = <font color="Maroon" title="c2">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E4:13"/>
<font color="Maroon" title="c3">K</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E2:13"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:13"/><b>then </b><i><font color="Green" title="E16">A8</font></i>: 
<font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">K</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/>
<a NAME="E6:13"/><b>then </b>
<font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">K</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a>, <a class="ref" href="tops_1.html#T48" title="TOPS_1:th.48">TOPS_1:48</a></i>;<br/>
<a NAME="E7:13"/><b>then </b><i><font color="Green" title="E17">A9</font></i>: 
<font color="Maroon" title="c3">K</font> is <a href="decomp_1.html#V2" title="DECOMP_1:attr.2">pre-open</a>
 
<b>by </b><i><a class="ref" href="decomp_1.html#D3" target="_self" title="DECOMP_1:def.3">Def3</a></i>;<br/>
<a NAME="E8:13"/><b>then </b><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c3">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K8" title="DECOMP_1:func.8">PO</a> <font color="Maroon" title="c1">T</font>
 
;<br/>
<a NAME="E9:13"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K2" title="DECOMP_1:func.2">pInt</a> <font color="Maroon" title="c3">K</font>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E5:13"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="decomp_1.html#E7:13"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="decomp_1.html#T4" target="_self" title="DECOMP_1:th.4">Th4</a></i>;<br/>
<a NAME="E10:13"/><b>then </b>
<font color="Maroon" title="c3">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> :  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K2" title="DECOMP_1:func.2">pInt</a> <font color="Olive" title="b1">B</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E11:13"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K8" title="DECOMP_1:func.8">PO</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K12" title="DECOMP_1:func.12">D(c,p)</a> <font color="Maroon" title="c1">T</font></span>)</span>
 <b>by </b><i><a class="txt" href="decomp_1.html#E8:13"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div>
