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

<span class="kw">let </span><font color="Maroon" title="c1">P</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> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r3</font>, <font color="Olive" title="b2">q3</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b3">r1</font>,<font color="Olive" title="b4">r2</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b3">r1</font>, <font color="Olive" title="b4">r2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Olive" title="b1">r3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">r1</font> &amp; <font color="Olive" title="b3">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">q3</font> ) </span>}</span>  holds <br/><font color="Maroon" title="c1">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>,  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> = <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">$1,$2</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">r3</font>, <font color="Maroon" title="c3">q3</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c2">r3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">q3</font> ) </span>}</span>  implies <font color="Maroon" title="c1">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>,  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>] <span class="kw">means </span><font color="Maroon" title="c2">r3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>,  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>] <span class="kw">means </span>$1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">q3</font>;<br/>
<a NAME="E1:83"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font>) where <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font>] </span>}</span>  is   <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> 2</span>)</span>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S9" title="DOMAIN_1:sch.9">DOMAIN_1:sch 9</a>();<br/></span>
<a NAME="E2:83"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font>) where <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font>] </span>}</span>  is   <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> 2</span>)</span>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S9" title="DOMAIN_1:sch.9">DOMAIN_1:sch 9</a>();<br/></span>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">Q1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c2">r3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r1</font> </span>}</span> , <font color="Maroon" title="c5">Q2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">q3</font> </span>}</span>  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> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E23">A1</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:83"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c2">r3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">q3</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Q2</font> ) &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Q2</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Q2</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:83_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Q2</font></span><br/><a NAME="E2:83_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c2">r3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">q3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>;<br/><a NAME="E3:83_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q1</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q2</font> )
 ;<br/><span class="kw">hence </span><a NAME="E4:83_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Q2</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:83_1"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Q2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font></span><br/><a NAME="E3:83_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q1</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="c7">r1</font>, <font color="Maroon" title="c8">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E5:83_1"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c7">r1</font>,<font color="Maroon" title="c8">r2</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c2">r3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r1</font> )
 ;<br/><a NAME="E6:83_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_1"><span class="lab"><font color="Green" title="E25">A3</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="c9">r19</font>, <font color="Maroon" title="c10">r29</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E8:83_1"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c9">r19</font>,<font color="Maroon" title="c10">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c9">r19</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">q3</font> )
 ;<br/><a NAME="E9:83_1"/>
( <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c7">r1</font>,<font color="Maroon" title="c8">r2</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r1</font> &amp; <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c9">r19</font>,<font color="Maroon" title="c10">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r19</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/><span class="kw">hence </span><a NAME="E10:83_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E5:83_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E8:83_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>;<br/>
<a NAME="E7:83"/>
( <font color="Maroon" title="c4">Q1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c5">Q2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1.html#T25" title="JORDAN1:th.25">JORDAN1:25</a>, <a class="ref" href="jordan1.html#T26" title="JORDAN1:th.26">JORDAN1:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:83"/>
<font color="Maroon" title="c1">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
