<?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="E1:21"/>
<span class="p1">(<span class="default"><a href="decomp_1.html#K10" title="DECOMP_1:func.10">PSO</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#K20" title="DECOMP_1:func.20">D(sp,ps)</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="decomp_1.html#K9" title="DECOMP_1:func.9">SPO</a> <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="21_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="E1:21_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#K10" title="DECOMP_1:func.10">PSO</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#K20" title="DECOMP_1:func.20">D(sp,ps)</a> <font color="Maroon" title="c1">T</font></span>)</span>
 ;<br/>

<a NAME="E2:21_1"/><b>then </b><i><font color="Green" title="E23">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#K10" title="DECOMP_1:func.10">PSO</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#K20" title="DECOMP_1:func.20">D(sp,ps)</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="E4:21_1"/><i><font color="Green" title="E24">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="E5:21_1"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c3">A</font> is <a href="decomp_1.html#V3" title="DECOMP_1:attr.3">pre-semi-open</a>
 ;<br/>
<a NAME="E6:21_1"/><i><font color="Green" title="E26">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#K4" title="DECOMP_1:func.4">psInt</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E5:21_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="decomp_1.html#T5" target="_self" title="DECOMP_1:th.5">Th5</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="E8:21_1"/><i><font color="Green" title="E27">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="E9:21_1"/><i><font color="Green" title="E28">A6</font></i>: 
 <a href="decomp_1.html#K5" title="DECOMP_1:func.5">spInt</a> <font color="Maroon" title="c4">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K4" title="DECOMP_1:func.4">psInt</a> <font color="Maroon" title="c4">Z</font>
 <b>by </b><i><a class="txt" href="decomp_1.html#E2:21_1"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<a NAME="E10:21_1"/>
<font color="Maroon" title="c4">Z</font> is <a href="decomp_1.html#V4" title="DECOMP_1:attr.4">semi-pre-open</a>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E4:21_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="decomp_1.html#E6:21_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="decomp_1.html#E8:21_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="decomp_1.html#E9:21_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="decomp_1.html#T6" target="_self" title="DECOMP_1:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E11:21_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K9" title="DECOMP_1:func.9">SPO</a> <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="decomp_1.html#E8:21_1"><i><font color="Green" title="E27">A5</font></i></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="E2:21"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K9" title="DECOMP_1:func.9">SPO</a> <font color="Maroon" title="c1">T</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">K</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="E4:21"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">K</font>
 <b>and </b><br/><a NAME="E5:21"/><i><font color="Green" title="E24">A8</font></i>: 
<font color="Maroon" title="c3">K</font> is <a href="decomp_1.html#V4" title="DECOMP_1:attr.4">semi-pre-open</a>
 ;<br/>
<a NAME="E6:21"/><i><font color="Green" title="E25">A9</font></i>: 
<font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<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></span>)</span>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E5:21"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="decomp_1.html#D5" target="_self" title="DECOMP_1:def.5">Def5</a></i>;<br/>
<a NAME="E7:21"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">K</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49" title="PRE_TOPC:th.49">PRE_TOPC:49</a>, <a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/>
<a NAME="E8:21"/><b>then </b>
 <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> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <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="tops_1.html#T48" title="TOPS_1:th.48">TOPS_1:48</a></i>;<br/>
<a NAME="E9:21"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<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></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49" title="PRE_TOPC:th.49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E10:21"/><b>then </b><i><font color="Green" title="E26">A10</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<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></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T58" title="TOPS_1:th.58">TOPS_1:58</a></i>;<br/>
<a NAME="E11:21"/>
 <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<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></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E12:21"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<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></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<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></span>)</span>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E10:21"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E13:21"/><b>then </b>
<font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<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></span>)</span>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E6:21"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E14:21"/><b>then </b><i><font color="Green" title="E27">A11</font></i>: 
<font color="Maroon" title="c3">K</font> is <a href="decomp_1.html#V3" title="DECOMP_1:attr.3">pre-semi-open</a>
 
<b>by </b><i><a class="ref" href="decomp_1.html#D4" target="_self" title="DECOMP_1:def.4">Def4</a></i>;<br/>
<a NAME="E15:21"/><b>then </b><i><font color="Green" title="E28">A12</font></i>: 
<font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K4" title="DECOMP_1:func.4">psInt</a> <font color="Maroon" title="c3">K</font>
 
<b>by </b><i><a class="ref" href="decomp_1.html#T5" target="_self" title="DECOMP_1:th.5">Th5</a></i>;<br/>
<a NAME="E16:21"/><i><font color="Green" title="E29">A13</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#K10" title="DECOMP_1:func.10">PSO</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E14:21"><i><font color="Green" title="E27">A11</font></i></a></i>;<br/>
<a NAME="E17:21"/>
 <a href="decomp_1.html#K5" title="DECOMP_1:func.5">spInt</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K4" title="DECOMP_1:func.4">psInt</a> <font color="Maroon" title="c3">K</font>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E5:21"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="decomp_1.html#E15:21"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="decomp_1.html#T6" target="_self" title="DECOMP_1:th.6">Th6</a></i>;<br/>
<a NAME="E18:21"/><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="decomp_1.html#K5" title="DECOMP_1:func.5">spInt</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K4" title="DECOMP_1:func.4">psInt</a> <font color="Olive" title="b1">B</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E19:21"/>
<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#K10" title="DECOMP_1:func.10">PSO</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#K20" title="DECOMP_1:func.20">D(sp,ps)</a> <font color="Maroon" title="c1">T</font></span>)</span>
 <b>by </b><i><a class="txt" href="decomp_1.html#E4:21"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="decomp_1.html#E16:21"><i><font color="Green" title="E29">A13</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div>
