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

<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="decomp_1.html#K9" title="DECOMP_1:func.9">SPO</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#K18" title="DECOMP_1:func.18">D(p,sp)</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K8" title="DECOMP_1:func.8">PO</a> <font color="Maroon" title="c1">T</font></span><br/>

<span class="kw">thus </span><a NAME="E1:14"/>
<span class="p1">(<span class="default"><a href="decomp_1.html#K9" title="DECOMP_1:func.9">SPO</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#K18" title="DECOMP_1:func.18">D(p,sp)</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#K8" title="DECOMP_1:func.8">PO</a> <font color="Maroon" title="c1">T</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="decomp_1.html#K8" title="DECOMP_1:func.8">PO</a> <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K9" title="DECOMP_1:func.9">SPO</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#K18" title="DECOMP_1:func.18">D(p,sp)</a> <font color="Maroon" title="c1">T</font></span>)</span></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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#K9" title="DECOMP_1:func.9">SPO</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#K18" title="DECOMP_1:func.18">D(p,sp)</a> <font color="Maroon" title="c1">T</font></span>)</span> or <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:14_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#K9" title="DECOMP_1:func.9">SPO</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#K18" title="DECOMP_1:func.18">D(p,sp)</a> <font color="Maroon" title="c1">T</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E2:14_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A0</font></span>: 
( <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> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K18" title="DECOMP_1:func.18">D(p,sp)</a> <font color="Maroon" title="c1">T</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E4:14_1"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c3">B</font> is  <a href="decomp_1.html#V4" title="DECOMP_1:attr.4">semi-pre-open</a>  )
 ;<br/>
<a NAME="E5:14_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K5" title="DECOMP_1:func.5">spInt</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="decomp_1.html#T6" target="_self" onmouseover="rs('decomp_1/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">B1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E7:14_1"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B1</font> &amp;  <a href="decomp_1.html#K2" title="DECOMP_1:func.2">pInt</a> <font color="Maroon" title="c4">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K5" title="DECOMP_1:func.5">spInt</a> <font color="Maroon" title="c4">B1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1"><span class="lab"><font color="Green" title="E16">A0</font></span></a></span>;<br/>
<a NAME="E8:14_1"/>
 <a href="decomp_1.html#K2" title="DECOMP_1:func.2">pInt</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E5:14_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E4:14_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a></span>;<br/>
<a NAME="E9:14_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">B</font> is  <a href="decomp_1.html#V2" title="DECOMP_1:attr.2">pre-open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="decomp_1.html#T4" target="_self" onmouseover="rs('decomp_1/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:14_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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#K9" title="DECOMP_1:func.9">SPO</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#K18" title="DECOMP_1:func.18">D(p,sp)</a> <font color="Maroon" title="c1">T</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:14"/>
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K9" title="DECOMP_1:func.9">SPO</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#K18" title="DECOMP_1:func.18">D(p,sp)</a> <font color="Maroon" title="c1">T</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:14"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">and </span><br/><a NAME="E5:14"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c3">K</font> is  <a href="decomp_1.html#V2" title="DECOMP_1:attr.2">pre-open</a> 
 ;<br/>
<a NAME="E6:14"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 <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> <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E7:14"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<a NAME="E8:14"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:14"><span class="lab"><font color="Green" title="E18">A3</font></span></a></span>;<br/>
<a NAME="E9:14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<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="E10:14"/><span class="kw">then </span>
<font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K5" title="DECOMP_1:func.5">spInt</a> <font color="Maroon" title="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="decomp_1.html#T6" target="_self" onmouseover="rs('decomp_1/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E11:14"/><span class="kw">then </span>
 <a href="decomp_1.html#K2" title="DECOMP_1:func.2">pInt</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K5" title="DECOMP_1:func.5">spInt</a> <font color="Maroon" title="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="decomp_1.html#T4" target="_self" onmouseover="rs('decomp_1/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E12:14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c3">K</font> <a href="tarski.html#R2" title="TARSKI: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#K2" title="DECOMP_1:func.2">pInt</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K5" title="DECOMP_1:func.5">spInt</a> <font color="Olive" title="b1">B</font> </span> } </span> 
 
;<br/>
<a NAME="E13:14"/>
<font color="Maroon" title="c3">K</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="decomp_1.html#K9" title="DECOMP_1:func.9">SPO</a> <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:14"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:14"/>
<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#K9" title="DECOMP_1:func.9">SPO</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#K18" title="DECOMP_1:func.18">D(p,sp)</a> <font color="Maroon" title="c1">T</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
