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<b>let </b><font color="Maroon">s1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">t1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">s2</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">t2</font> be   <a href="real_1.html#NM1">Real</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>, <font color="Maroon">P1</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>, <font color="Maroon">P2</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><b>that </b><br/><a NAME="E1:46"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t2</font> )
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green">E30</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="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t1</font> ) or ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s1</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> ) or ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s1</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> ) or ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t1</font> ) ) </span>}</span> 
 <b>and </b><br/><a NAME="E3:46"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">pa</font> where <font color="Olive">pa</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">s1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">pa</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">pa</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">pa</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">pa</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t2</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:46"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">pc</font> where <font color="Olive">pc</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> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">pc</font> <a href="euclid.html#K21">`1</a>  or  not <font color="Olive">pc</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">pc</font> <a href="euclid.html#K22">`2</a>  or  not <font color="Olive">pc</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) </span>}</span> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:46_1"/><i><font color="Green">E33</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="E2:46_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E3:46_1"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <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">pd</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><a class="ref" href="jordan1.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E5:46_1"/>
( not <!--HUMANRECFAILED-->( <font color="Maroon">pd</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">pd</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">pd</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t1</font> ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">pd</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">pd</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">pd</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">pd</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">pd</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">pd</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">pd</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">pd</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">pd</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t1</font> ) )
 <b>by </b><i><a class="txt" href="jordan1.html#E3"><i><font color="Green">Lemma48</font></i></a>, </i>;<br/><a NAME="E6:46_1"/><b>then </b>
( ( <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">pd</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">pd</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">pd</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">pd</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t2</font> ) or  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pd</font> <a href="euclid.html#K21">`1</a>  or  not <font color="Maroon">pd</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pd</font> <a href="euclid.html#K22">`2</a>  or  not <font color="Maroon">pd</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> )
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E7:46_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> )
 <b>by </b><i><a class="ref" href="jordan1.html#T4" target="_self">Th4</a>, </i>;<br/><b>hence </b><a NAME="E8:46_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:46"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:46_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1_1"><b>suppose </b></a><a NAME="E1:46_2_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 ;<br/><div class="add"><a NAME="E2:46_2_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 ex <font color="Olive">pa</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> st <br/>( <font color="Olive">pa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">pa</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">pa</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">pa</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">pa</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t2</font> )
 <b>by </b><i><a class="ref" href="jordan1.html#T4" target="_self">Th4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:46_2_1_1_1_1"/><b>then </b>
 ex <font color="Olive">pb</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> st <br/>( <font color="Olive">pb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( ( <font color="Olive">pb</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <font color="Olive">pb</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> &amp; <font color="Olive">pb</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t1</font> ) or ( <font color="Olive">pb</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Olive">pb</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s1</font> &amp; <font color="Olive">pb</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> ) or ( <font color="Olive">pb</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Olive">pb</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s1</font> &amp; <font color="Olive">pb</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> ) or ( <font color="Olive">pb</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> &amp; <font color="Olive">pb</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> &amp; <font color="Olive">pb</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t1</font> ) ) )
 <b>by </b><i><a class="txt" href="jordan1.html#E3"><i><font color="Green">Lemma48</font></i></a></i>;<br/><b>hence </b><a NAME="E3:46_2_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:46_2_1_1_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> 
 <b>by </b><i>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1_2"><b>suppose </b></a><a NAME="E1:46_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">pc</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="E3:46_2_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">pc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:46_2_1_1_2"/><i><font color="Green">E42</font></i>: 
(  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pc</font> <a href="euclid.html#K21">`1</a>  or  not <font color="Maroon">pc</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pc</font> <a href="euclid.html#K22">`2</a>  or  not <font color="Maroon">pc</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_2_1_1_2_1"/>
<font color="Maroon">pc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:46_2_1_1_2_1"/><b>then </b>
 ex <font color="Olive">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> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pc</font> &amp; ( ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t1</font> ) or ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s1</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> ) or ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s1</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> ) or ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t1</font> ) ) )
 <b>by </b><i><a class="txt" href="jordan1.html#E3"><i><font color="Green">Lemma48</font></i></a></i>;<br/><b>hence </b><a NAME="E3:46_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1.html#E15:46"><i><font color="Green">E47</font></i></a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:46_2_1_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#T50">SUBSET_1:50</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:46_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> 
 ;<br/></div><b>end;</b></div>
<a NAME="E8:46"/><b>then </b>
<font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:46"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font>
 <b>by </b><i><a class="ref" href="jordan1.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>

<b>set </b><font color="Maroon">s3</font> = <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s2</font></span>)</span> <a href="real_1.html#K6">/</a> 2;<br/>
<b>set </b><font color="Maroon">t3</font> = <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2;<br/>
<a NAME="E10:46"/>
( <font color="Maroon">s1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font> )
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:46"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E11:46"/><b>then </b><i><font color="Green">E44</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s1</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t1</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E12:46"/>
( <font color="Maroon">s1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s2</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t2</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font> )
 
<b>by </b><i><a class="txt" href="jordan1.html#E15:46"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E13:46"/><b>then </b><i><font color="Green">E45</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t2</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>set </b><font color="Maroon">pp</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s2</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>
<a NAME="E14:46"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s2</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s2</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E15:46"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s2</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">pp2</font> where <font color="Olive">pp2</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">s1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">pp2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">pp2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">pp2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">pp2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t2</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:46"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 <b>by </b><i><a class="ref" href="jordan1.html#T4" target="_self">Th4</a></i>;<br/>
<b>thus </b><a NAME="E18:46"/>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>

<b>set </b><font color="Maroon">P'</font> = <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> ;<br/>
<a NAME="E19:46"/>
<font color="Maroon">P1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P2</font>
 
<b>by </b><i><a class="ref" href="jordan1.html#T4" target="_self">Th4</a>, , <a class="txt" href="jordan1.html#E13:46"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>hence </b><a NAME="E20:46"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P1A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>, <font color="Maroon">P2B</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>;<br/><b>assume </b><a NAME="E1:46_3"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">P1A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">P2B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font> )
 ;<br/><a NAME="E2:46_3"/>
<font color="Maroon">P1</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan1.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E3:46_3"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">P1A</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/><a NAME="E4:46_3"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">P2</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i>, </i>;<br/><a NAME="E5:46_3"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">sa</font>,<font color="Olive">ta</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">sa</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">ta</font> is   <a href="real_1.html#NM1">Real</a> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">sa</font> or  not <font color="Olive">sa</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">ta</font> or  not <font color="Olive">ta</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">A0</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s3a</font>,<font color="Olive">t3a</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s3a</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t3a</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s3a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> </span>}</span>  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> <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">A1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s4</font>,<font color="Olive">t4</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s4</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t4</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">t4</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t1</font> </span>}</span>  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> <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">A2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s5</font>,<font color="Olive">t5</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s5</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t5</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s5</font> </span>}</span>  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> <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">A3</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s6</font>,<font color="Olive">t6</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s6</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t6</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">t6</font> </span>}</span>  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> <b>by </b><i>, </i>;<br/><a NAME="E10:46_3"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A3</font>
 <b>by </b><i>, </i>;<br/><a NAME="E11:46_3"/>
<font color="Maroon">t2</font> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">t2</font>
 <b>by </b><i><a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/><a NAME="E12:46_3"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="real_1.html#K3">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">t2</font> <a href="real_1.html#K3">+</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A3</font>
 ;<br/><a NAME="E13:46_3"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">P2B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="jordan1.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/><a NAME="E14:46_3"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">P2B</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i>, <a class="ref" href="jordan1.html#T6" target="_self">Th6</a>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/><a NAME="E15:46_3"/><i><font color="Green">E173</font></i>: 
for <font color="Olive">CP</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>  st <font color="Olive">CP</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">P1A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">CP</font> holds <br/><font color="Maroon">P1A</font> <a href="hidden.html#R1">=</a> <font color="Olive">CP</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">CP</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:46_3_1"/>
<font color="Maroon">CP</font> is <a href="connsp_1.html#V2">connected</a>
 ;<br/>

<a NAME="E2:46_3_1"/><b>then </b><i><font color="Green">E175</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font> is <a href="connsp_1.html#V1">connected</a>
 
<b>by </b><i><a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_3_1_1"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">P1A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">CP</font>
 ;<br/><a NAME="E2:46_3_1_1"/>
<font color="Maroon">P1A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:46_3_1_1_1"/>
<font color="Maroon">P1A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">CP</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E2:46_3_1_1_1"/>
<font color="Maroon">P1A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">AP</font> = <font color="Maroon">P1A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span> ;<br/><a NAME="E4:46_3_1_1"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1A</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>;<br/><a NAME="E5:46_3_1_1"/>
<font color="Maroon">P1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="jordan1.html#T4" target="_self">Th4</a>, <a class="txt" href="jordan1.html#E9:46"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E6:46_3_1_1"/><b>then </b><i><font color="Green">E179</font></i>: 
<font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E7:46_3_1_1"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E8:46_3_1_1"/>
<font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1A</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E9:46_3_1_1"/><b>then </b><i><font color="Green">E181</font></i>: 
<font color="Maroon">P1A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/><a NAME="E10:46_3_1_1"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">CP</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E11:46_3_1_1"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">AP</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan1.html#T4" target="_self">Th4</a>, , , , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E12:46_3_1_1"/>
<font color="Maroon">AP</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan1.html#T9" target="_self">Th9</a>, , <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/><a NAME="E13:46_3_1_1"/><b>then </b><i><font color="Green">E184</font></i>: 
<font color="Maroon">AP</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E14:46_3_1_1"/>
<font color="Maroon">P2B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:46_3_1_1_2"/>
<font color="Maroon">P2B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">CP</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E2:46_3_1_1_2"/>
<font color="Maroon">P2B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">BP</font> = <font color="Maroon">P2B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span> ;<br/><a NAME="E16:46_3_1_1"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P2B</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>;<br/><a NAME="E17:46_3_1_1"/>
<font color="Maroon">P2</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="txt" href="jordan1.html#E11:46"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E18:46_3_1_1"/><b>then </b><i><font color="Green">E187</font></i>: 
<font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E19:46_3_1_1"/>
<font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P2B</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E20:46_3_1_1"/><b>then </b><i><font color="Green">E188</font></i>: 
<font color="Maroon">P2B</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/><a NAME="E21:46_3_1_1"/>
<font color="Maroon">CP</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E22:46_3_1_1"/><b>then </b>
<font color="Maroon">BP</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/><a NAME="E23:46_3_1_1"/><b>then </b><i><font color="Green">E189</font></i>: 
<font color="Maroon">BP</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E24:46_3_1_1"/>
<font color="Maroon">CP</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i/>;<br/><b>then </b><i><font color="Green">E190</font></i>: <font color="Maroon">CP</font> = 
<span class="p1">(<span class="default"><font color="Maroon">P1A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font>
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">AP</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">BP</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_3_1_1_4"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">BP</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><i><font color="Green">E192</font></i>: <font color="Maroon">AP</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">BP</font> = 
<span class="p1">(<span class="default"><font color="Maroon">P1A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">P2B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P1A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P2B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P1A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P2B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">CP</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P1A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P2B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font>

;<br/><a NAME="E3:46_3_1_1_4"/>
<font color="Maroon">P1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P2</font>
 <b>by </b><i><a class="ref" href="jordan1.html#T4" target="_self">Th4</a>, , <a class="txt" href="jordan1.html#E13:46"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E4:46_3_1_1_4"/><b>then </b>
<font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E5:46_3_1_1_4"/><b>then </b>
<font color="Maroon">AP</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">BP</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E6:46_3_1_1_4"/>
contradiction
 <b>by </b><i>, , , , , , , <a class="ref" href="connsp_1.html#T12">CONNSP_1:12</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E27:46_3_1_1"/>
(  not <font color="Maroon">P1A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">CP</font> or <font color="Maroon">P1A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CP</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:46_3_1"/>
(  not <font color="Maroon">P1A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">CP</font> or <font color="Maroon">P1A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CP</font> )
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E16:46_3"/>
<font color="Maroon">P1A</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/><a NAME="E17:46_3"/>
for <font color="Olive">CP</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>  st <font color="Olive">CP</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">P2B</font> <a href="tarski.html#R1">c=</a> <font color="Olive">CP</font> holds <br/><font color="Maroon">P2B</font> <a href="hidden.html#R1">=</a> <font color="Olive">CP</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">CP</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:46_3_2"/>
<font color="Maroon">CP</font> is <a href="connsp_1.html#V2">connected</a>
 ;<br/>

<a NAME="E2:46_3_2"/><b>then </b><i><font color="Green">E193</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font> is <a href="connsp_1.html#V1">connected</a>
 
<b>by </b><i><a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/>
<b>assume </b><a NAME="E3:46_3_2"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">P2B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">CP</font>
 ;<br/>

<a NAME="E4:46_3_2"/>
<font color="Maroon">P2B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:46_3_2_1"/>
<font color="Maroon">P2B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">CP</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E2:46_3_2_1"/>
<font color="Maroon">P2B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">BP</font> = <font color="Maroon">P2B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span> ;<br/>
<a NAME="E6:46_3_2"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P2B</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>;<br/>
<a NAME="E7:46_3_2"/>
<font color="Maroon">P2</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="txt" href="jordan1.html#E11:46"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E8:46_3_2"/><b>then </b><i><font color="Green">E196</font></i>: 
<font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E9:46_3_2"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E10:46_3_2"/>
<font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P2B</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E11:46_3_2"/><b>then </b><i><font color="Green">E198</font></i>: 
<font color="Maroon">P2B</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E12:46_3_2"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">CP</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E13:46_3_2"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">BP</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1.html#T8" target="_self">Th8</a>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E14:46_3_2"/>
<font color="Maroon">BP</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1.html#T9" target="_self">Th9</a>, , <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E15:46_3_2"/><b>then </b><i><font color="Green">E201</font></i>: 
<font color="Maroon">BP</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E16:46_3_2"/>
<font color="Maroon">P1A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:46_3_2_2"/>
<font color="Maroon">P1A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">CP</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E2:46_3_2_2"/>
<font color="Maroon">P1A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">AP</font> = <font color="Maroon">P1A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span> ;<br/>
<a NAME="E18:46_3_2"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1A</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>;<br/>
<a NAME="E19:46_3_2"/>
<font color="Maroon">P1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="jordan1.html#T4" target="_self">Th4</a>, <a class="txt" href="jordan1.html#E9:46"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E20:46_3_2"/><b>then </b><i><font color="Green">E203</font></i>: 
<font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E21:46_3_2"/>
<font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1A</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E22:46_3_2"/><b>then </b><i><font color="Green">E204</font></i>: 
<font color="Maroon">P1A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E23:46_3_2"/>
<font color="Maroon">CP</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E24:46_3_2"/><b>then </b>
<font color="Maroon">AP</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">CP</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E25:46_3_2"/><b>then </b><i><font color="Green">E205</font></i>: 
<font color="Maroon">AP</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E26:46_3_2"/>
<font color="Maroon">CP</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E206</font></i>: <font color="Maroon">CP</font> = 
<span class="p1">(<span class="default"><font color="Maroon">P1A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font>
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">AP</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">BP</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_3_2_4"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">AP</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><font color="Maroon">AP</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">BP</font> = 
<span class="p1">(<span class="default"><font color="Maroon">P1A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">P2B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P1A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P2B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P1A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P2B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">CP</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P1A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P2B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CP</font>

;<br/><a NAME="E3:46_3_2_4"/><b>then </b>
<font color="Maroon">AP</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">BP</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:46_3_2_4"/>
contradiction
 <b>by </b><i>, , , , , , , <a class="ref" href="connsp_1.html#T12">CONNSP_1:12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E29:46_3_2"/>
<font color="Maroon">P2B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CP</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E18:46_3"/>
( <font color="Maroon">P1A</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Maroon">P2B</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E22:46"/>
for <font color="Olive">P1A</font>, <font color="Olive">P2B</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>  st <font color="Olive">P1A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> &amp; <font color="Olive">P2B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font> holds <br/>( <font color="Olive">P1A</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">P2B</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> )
 ;<br/>


</div>
