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

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:5"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> </span>}</span> 
 ;<br/>

<a NAME="E2:5"/><i><font color="Green">E47</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:5"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<a NAME="E1:5_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:5_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E2:5_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/>
<a NAME="E3:5_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E5:5_1_1"/>
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:5_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:5_1"/>
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:5_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:5_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:5_1_2_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 ;<br/><b>thus </b><a NAME="E4:5_1_2_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E5:5_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">P</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:5_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:5_1"/>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:5"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan2b.html#K1">Proj</a> <font color="Olive">p</font>,1 </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<a NAME="E1:5_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan2b.html#K1">Proj</a> <font color="Olive">p</font>,1 </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:5_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:5_2_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:5_2_1"/>
 <a href="jordan2b.html#K1">Proj</a> <font color="Maroon">p</font>,1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="jordan2b.html#T35">JORDAN2B:35</a></i>;<br/>
<b>hence </b><a NAME="E5:5_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan2b.html#K1">Proj</a> <font color="Olive">p</font>,1 </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:5_2"/>
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan2b.html#K1">Proj</a> <font color="Olive">p</font>,1 </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:5_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan2b.html#K1">Proj</a> <font color="Olive">p</font>,1 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:5_2_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1 )
 ;<br/>
<a NAME="E4:5_2_2"/>
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="jordan2b.html#T35">JORDAN2B:35</a></i>;<br/>
<b>hence </b><a NAME="E5:5_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:5_2"/>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan2b.html#K1">Proj</a> <font color="Olive">p</font>,1 </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">P1</font> = <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E6:5"/>
<font color="Maroon">P1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, , <a class="ref" href="jordan2b.html#T17">JORDAN2B:17</a></i>;<br/>
<b>hence </b><a NAME="E7:5"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>


</div>
