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

<b>let </b><font color="Maroon" title="c1">s1</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">P2</font> being  <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>  st <font color="Olive" title="b1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b3">r</font>,<font color="Olive" title="b4">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b3">r</font>, <font color="Olive" title="b4">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b4">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">s1</font> </span>}</span>  &amp; <font color="Olive" title="b2">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b3">r2</font>,<font color="Olive" title="b4">s2</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b3">r2</font>, <font color="Olive" title="b4">s2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b4">s2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">s1</font> </span>}</span>  holds <br/><font color="Olive" title="b1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span><br/><b>let </b><font color="Maroon" title="c2">P1</font>, <font color="Maroon" title="c3">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> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">s1</font> </span>}</span>  &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r2</font>,<font color="Olive" title="b2">s2</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r2</font>, <font color="Olive" title="b2">s2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b2">s2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">s1</font> </span>}</span>  implies <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">s1</font> </span>}</span>  &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r2</font>,<font color="Olive" title="b2">s2</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r2</font>, <font color="Olive" title="b2">s2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b2">s2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">s1</font> </span>}</span>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span><br/>

<a NAME="E2:17"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c2">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/>

<b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span><br/>

<b>then consider </b><font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c6">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:17_1"/><i><font color="Green" title="E16">A3</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c6">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">s1</font> )
 <b>by </b><i><a class="txt" href="gobrd11.html#E1:17"><i><font color="Green" title="E14">A1</font></i></a></i>;<br/>
<a NAME="E4:17_1"/>
for <font color="Olive" title="b1">r2</font>, <font color="Olive" title="b2">s2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  holds <br/>(  not <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r2</font>,<font color="Olive" title="b2">s2</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> or  not <font color="Olive" title="b2">s2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">s1</font> )
 
<b>by </b><i><a class="txt" href="gobrd11.html#E3:17_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="sppol_2.html#T1" title="SPPOL_2:th.1">SPPOL_2:1</a></i>;<br/>
<a NAME="E5:17_1"/><b>then </b>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2 &amp; not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font> )
 
<b>by </b><i><a class="txt" href="gobrd11.html#E1:17"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="gobrd11.html#E3:17_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="gobrd11.html#E8"><i><font color="Green" title="E4">Lm2</font></i></a></i>;<br/>
<a NAME="E6:17_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">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="E7:17_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">P2</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/>
<b>hence </b><a NAME="E8:17_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#D5" title="SUBSET_1:def.5">SUBSET_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:17"/>
<font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font> )</span><br/>

<b>assume </b><a NAME="E1:17_2"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font></span><br/>

<a NAME="E2:17_2"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">P2</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#D5" title="SUBSET_1:def.5">SUBSET_1:def 5</a></i>;<br/>
<a NAME="E3:17_2"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">P2</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/>
<a NAME="E4:17_2"/><b>then </b><i><font color="Green" title="E17">A5</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2 &amp; not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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/>
<b>reconsider </b><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c4">x</font> as   <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> 2</span>)</span> <b>by </b><i><a class="txt" href="gobrd11.html#E1:17_2"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/>
<a NAME="E6:17_2"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<a NAME="E7:17_2"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">s1</font>
 
<b>by </b><i><a class="txt" href="gobrd11.html#E1:17"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="gobrd11.html#E4:17_2"><i><font color="Green" title="E17">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:17_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font>
 <b>by </b><i><a class="txt" href="gobrd11.html#E1:17"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="gobrd11.html#E6:17_2"><i><font color="Green" title="E18">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:17"/>
<font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="gobrd11.html#E2:17"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
