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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c3">P1</font>, <font color="Maroon" title="c4">P2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c3">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> &amp; <font color="Maroon" title="c4">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> &amp; <font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">P21</font> = <font color="Maroon" title="c4">P2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<a NAME="E3:42"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c8">P21</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="jordan6.html#E1:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="jordan5a.html#T1" title="JORDAN5A:th.1">JORDAN5A:1</a></i>;<br/>
<a NAME="E4:42"/>
<font color="Maroon" title="c6">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">P2</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E1:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:42"/><b>then </b><i><font color="Green" title="E28">A3</font></i>: 
( <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P1</font> &amp; <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P2</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/>
<a NAME="E6:42"/><i><font color="Green" title="E29">A4</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E7:42"/><b>then </b><i><font color="Green" title="E30">A5</font></i>: 
not <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 
<b>by </b><i><a class="txt" href="jordan6.html#E1:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">P2'</font> = <font color="Maroon" title="c8">P21</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font></span>)</span> <b>by </b><i><a class="txt" href="jordan6.html#E1:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="jordan6.html#E6:42"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E9:42"/>
<font color="Maroon" title="c7">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">P2</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E1:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E10:42"/><b>then </b><i><font color="Green" title="E31">A6</font></i>: 
( <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P1</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P2</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/>
<a NAME="E11:42"/><i><font color="Green" title="E32">A7</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font> <a href="compts_1.html#NR2" title="COMPTS_1:NR.2">is_T2</a> 
 
<b>by </b><i><a class="txt" href="jordan6.html#E7:42"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="topmetr.html#T3" title="TOPMETR:th.3">TOPMETR:3</a></i>;<br/>
<a NAME="E12:42"/>
<font color="Maroon" title="c9">P2'</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="jordan6.html#E3:42"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="jordan6.html#E6:42"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="compts_1.html#T11" title="COMPTS_1:th.11">COMPTS_1:11</a></i>;<br/>
<a NAME="E13:42"/><b>then </b><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c9">P2'</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="jordan6.html#E11:42"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="compts_1.html#T16" title="COMPTS_1:th.16">COMPTS_1:16</a></i>;<br/>
<i><font color="Green" title="E34">A9</font></i>: <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">P2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">P2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">P2</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan6.html#E1:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="xboole_1.html#T42" title="XBOOLE_1:th.42">XBOOLE_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">P2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>
<br/>.= 
<font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">P2</font>

;<br/>
<a NAME="E15:42"/>
<font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<a NAME="E1:42_2"/><i><font color="Green" title="E35">A10</font></i>: 
<font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">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="E1:42_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">P2</font>
 ;<br/>

<a NAME="E2:42_2_1"/><b>then </b><i><font color="Green" title="E36">A11</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P1</font> &amp; not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:42_2_1"/><b>then </b>
not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan6.html#E5:42"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="jordan6.html#E10:42"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:42_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E1:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="jordan6.html#E2:42_2_1"><i><font color="Green" title="E36">A11</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:42_2"/>
<font color="Maroon" title="c5">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">P2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">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="E1:42_2_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font>
 ;<br/>

<a NAME="E2:42_2_2"/><b>then </b><i><font color="Green" title="E36">A12</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P1</font> &amp; not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan6.html#E1:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:42_2_2"/><b>then </b>
not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P2</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E1:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:42_2_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">P2</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E2:42_2_2"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:42_2"/>
<font color="Maroon" title="c3">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E1:42_2"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:42"/><b>then </b>
<font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">P2'</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="jordan6.html#E6:42"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="jordan6.html#E14:42"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="subset_1.html#D5" title="SUBSET_1:def.5">SUBSET_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E17:42"/>
<font color="Maroon" title="c5">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="jordan6.html#E13:42"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/>


</div>
