<?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" 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#K7" title="DECOMP_1:func.7">SO</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#K16" title="DECOMP_1:func.16">D(alpha,s)</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a> </span><br/>

<span class="kw">thus </span><a NAME="E1:16"/>
<span class="p1">(<span class="default"><a href="decomp_1.html#K7" title="DECOMP_1:func.7">SO</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#K16" title="DECOMP_1:func.16">D(alpha,s)</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">T</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">T</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K7" title="DECOMP_1:func.7">SO</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#K16" title="DECOMP_1:func.16">D(alpha,s)</a> <font color="Maroon" title="c1">T</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_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">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" 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#K7" title="DECOMP_1:func.7">SO</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#K16" title="DECOMP_1:func.16">D(alpha,s)</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> <font color="Maroon" title="c1">T</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_1"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<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#K7" title="DECOMP_1:func.7">SO</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#K16" title="DECOMP_1:func.16">D(alpha,s)</a> <font color="Maroon" title="c1">T</font></span>)</span>
 ; <a class="txt" 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> <font color="Maroon" title="c1">T</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a> </span><br/>

<a NAME="E2:16_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K7" title="DECOMP_1:func.7">SO</a> <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:16_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="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E5:16_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c3">A</font> is  <a href="decomp_1.html#V1" title="DECOMP_1:attr.1">semi-open</a> 
 ;<br/>
<a NAME="E6:16_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K16" title="DECOMP_1:func.16">D(alpha,s)</a> <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E8:16_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Z</font>
 <span class="kw">and </span><br/><a NAME="E9:16_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
 <a href="decomp_1.html#K3" title="DECOMP_1:func.3">alphaInt</a> <font color="Maroon" title="c4">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K1" title="DECOMP_1:func.1">sInt</a> <font color="Maroon" title="c4">Z</font>
 ;<br/>
<a NAME="E10:16_1"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K1" title="DECOMP_1:func.1">sInt</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="decomp_1.html#T3" target="_self" title="DECOMP_1:th.3">Th3</a></span>;<br/>
<a NAME="E11:16_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">Z</font> is    <a href="decomp_1.html#M1" title="DECOMP_1:mode.1">alpha-set</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E8:16_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E9:16_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="decomp_1.html#T2" target="_self" title="DECOMP_1:th.2">Th2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:16_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a></span>; <a class="txt" 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">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" 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> <font color="Maroon" title="c1">T</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a>  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#K7" title="DECOMP_1:func.7">SO</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#K16" title="DECOMP_1:func.16">D(alpha,s)</a> <font color="Maroon" title="c1">T</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:16"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a> 
 ; <a class="txt" 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#K7" title="DECOMP_1:func.7">SO</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#K16" title="DECOMP_1:func.16">D(alpha,s)</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="E4:16"/><span class="lab"><font color="Green" title="E18">A6</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="E5:16"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
<font color="Maroon" title="c3">K</font> is    <a href="decomp_1.html#M1" title="DECOMP_1:mode.1">alpha-set</a> of <font color="Maroon" title="c1">T</font>
 ;<br/>
<a NAME="E6:16"/><span class="lab"><font color="Green" title="E20">A8</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> <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="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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T16" title="TOPS_1:th.16">TOPS_1:16</a></span>;<br/>
<a NAME="E7:16"/>
<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> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="decomp_1.html#D1" target="_self" title="DECOMP_1:def.1">Def1</a></span>;<br/>
<a NAME="E8:16"/><span class="kw">then </span>
<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> <font color="Maroon" title="c3">K</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E9:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A9</font></span>: 
<font color="Maroon" title="c3">K</font> is  <a href="decomp_1.html#V1" title="DECOMP_1:attr.1">semi-open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="decomp_1.html#D2" target="_self" title="DECOMP_1:def.2">Def2</a></span>;<br/>
<a NAME="E10:16"/><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#K1" title="DECOMP_1:func.1">sInt</a> <font color="Maroon" title="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="decomp_1.html#T3" target="_self" title="DECOMP_1:th.3">Th3</a></span>;<br/>
<a NAME="E11:16"/><span class="kw">then </span>
 <a href="decomp_1.html#K3" title="DECOMP_1:func.3">alphaInt</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K1" title="DECOMP_1:func.1">sInt</a> <font color="Maroon" title="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="decomp_1.html#T2" target="_self" title="DECOMP_1:th.2">Th2</a></span>;<br/>
<a NAME="E12:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<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#K3" title="DECOMP_1:func.3">alphaInt</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="decomp_1.html#K1" title="DECOMP_1:func.1">sInt</a> <font color="Olive" title="b1">B</font> </span> } </span> 
 
;<br/>
<a NAME="E13:16"/>
<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> : <font color="Olive" title="b1">B</font> is  <a href="decomp_1.html#V1" title="DECOMP_1:attr.1">semi-open</a>  </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E21">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:16"/>
<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#K7" title="DECOMP_1:func.7">SO</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#K16" title="DECOMP_1:func.16">D(alpha,s)</a> <font color="Maroon" title="c1">T</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
