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

<b>let </b><font color="Maroon">P</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">P1</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">P2</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">P1'</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">P2'</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p1</font> be   <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">p2</font> be   <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>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:53"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">P</font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a> 
 <b>and </b><br/><a NAME="E2:53"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">P1</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E3:53"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">P2</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E4:53"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E5:53"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">P1'</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E6:53"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">P2'</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E7:53"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">P1'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="topreal2.html#NM1">Simple_closed_curve</a> <b>by </b><i/>;<br/>
<a NAME="E9:53"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:53"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">P2'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E11:53"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1'</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1'</font> )
 
<b>by </b><i><a class="ref" href="jordan6.html#T4" target="_self">Th4</a>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E12:53"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">P1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E13:53"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> )
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E14:53"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> )
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E15:53"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E16:53"/>
 <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> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E17:53"/><b>then </b><i><font color="Green">E80</font></i>: 
the <a href="struct_0.html#U1">carrier</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> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
;<br/>
<a NAME="E18:53"/>
<font color="Maroon">P1</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Q1</font> = <font color="Maroon">P1</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> as   <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> <font color="Maroon">P</font></span>)</span> <b>by </b><i><a class="txt" href="jordan6.html#E5:53"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan6.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E20:53"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">Q1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E21:53"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">P2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E22:53"/>
<font color="Maroon">P2</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Q2</font> = <font color="Maroon">P2</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> as   <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> <font color="Maroon">P</font></span>)</span> <b>by </b><i><a class="ref" href="jordan6.html#T5" target="_self">Th5</a>, <a class="txt" href="jordan6.html#E7:53"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E24:53"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">Q2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E25:53"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">P1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E26:53"/>
<font color="Maroon">P1'</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Q1'</font> = <font color="Maroon">P1'</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> as   <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> <font color="Maroon">P</font></span>)</span> <b>by </b><i><a class="ref" href="jordan6.html#T5" target="_self">Th5</a>, <a class="txt" href="jordan6.html#E10:53"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E28:53"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">Q1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E29:53"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">P2'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E30:53"/>
<font color="Maroon">P2'</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Q2'</font> = <font color="Maroon">P2'</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> as   <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> <font color="Maroon">P</font></span>)</span> <b>by </b><i><a class="ref" href="jordan6.html#T5" target="_self">Th5</a>, <a class="txt" href="jordan6.html#E12:53"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E32:53"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">Q2'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E33:53"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">Q1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T42">XBOOLE_1:42</a></i>;<br/>
<a NAME="E34:53"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">Q1'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T42">XBOOLE_1:42</a></i>;<br/>
<a NAME="E35:53"/><b>then </b>
<font color="Maroon">Q1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E36:53"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">Q1'</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Q1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E37:53"/>
<font color="Maroon">Q2'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E38:53"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">Q2'</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Q1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E39:53"/>
<font color="Maroon">Q1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q1'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2'</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E40:53"/><b>then </b><i><font color="Green">E148</font></i>: 
<font color="Maroon">Q1</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Q1'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E41:53"/>
<font color="Maroon">Q2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q1'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2'</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E42:53"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">Q2</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Q1'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E43:53"/>
 <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> <font color="Maroon">P1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">R1</font> = <font color="Maroon">Q1</font> as   <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> <font color="Maroon">P1</font></span>)</span> <b>by </b><i><a class="txt" href="jordan6.html#E6:53"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E45:53"/>
<font color="Maroon">R1</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E46:53"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">Q1</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="jordan6.html#E5:53"><i><font color="Green">E52</font></i></a>, </i>;<br/>
<a NAME="E47:53"/>
 <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> <font color="Maroon">P2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">R2</font> = <font color="Maroon">Q2</font> as   <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> <font color="Maroon">P2</font></span>)</span> <b>by </b><i><a class="txt" href="jordan6.html#E9:53"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E49:53"/>
<font color="Maroon">R2</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E50:53"/><b>then </b><i><font color="Green">E155</font></i>: 
<font color="Maroon">Q2</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="jordan6.html#E7:53"><i><font color="Green">E55</font></i></a>, </i>;<br/>
<a NAME="E51:53"/>
 <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> <font color="Maroon">P1'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">R1'</font> = <font color="Maroon">Q1'</font> as   <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> <font color="Maroon">P1'</font></span>)</span> <b>by </b><i><a class="txt" href="jordan6.html#E11:53"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E53:53"/>
<font color="Maroon">R1'</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="ref" href="jordan6.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E54:53"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">Q1'</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="jordan6.html#E3:53"><i><font color="Green">E49</font></i></a>, </i>;<br/>
<a NAME="E55:53"/>
 <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> <font color="Maroon">P2'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">R2'</font> = <font color="Maroon">Q2'</font> as   <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> <font color="Maroon">P2'</font></span>)</span> <b>by </b><i><a class="txt" href="jordan6.html#E13:53"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E57:53"/>
<font color="Maroon">R2'</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E58:53"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">Q2'</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="jordan6.html#E2:53"><i><font color="Green">E47</font></i></a>, </i>;<br/>
<a NAME="E59:53"/><i><font color="Green">E159</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:53_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:53_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E4:53"><i><font color="Green">E51</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E60:53"/><i><font color="Green">E161</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:53_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:53_2"/>
<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="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E61:53"/><i><font color="Green">E162</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:53_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:53_3"/>
<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="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E62:53"/><i><font color="Green">E164</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:53_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<a NAME="E2:53_4"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:53_4"/>
<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="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E63:53"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">Q1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E64:53"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">Q2</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T12" target="_self">Th12</a>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E65:53"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">Q1'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E66:53"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">Q2'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T13" target="_self">Th13</a>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_5"/><i><font color="Green">E170</font></i>: 
(  not <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font> or  not <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_5_1_1"/>
( <font color="Maroon">P1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P2'</font> or <font color="Maroon">P2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P1'</font> )
 </span><b>by </b><i><a class="txt" href="jordan6.html#E15:53"><i><font color="Green">E79</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1_1_1"><b>case </b></a><a NAME="E1:53_5_1_1_1"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">P1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P2'</font>
 ;<br/><div class="add"><div><i><font color="Green">E172</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_5_1_1_1_1"/><i><font color="Green">E173</font></i>: 
( <font color="Maroon">Q1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q2'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q2'</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><a NAME="E2:53_5_1_1_1_1"/><b>then </b><i><font color="Green">E175</font></i>: 
<font color="Maroon">Q1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q2'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:53_5_1_1_1_1"/>
<font color="Maroon">Q2'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q1</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E21:53"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>hence </b><a NAME="E4:53_5_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan6.html#E14:53"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan6.html#E17:53"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan6.html#E24:53"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_5_1_1_1_2_1"/>
( <font color="Maroon">Q1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q2'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Q2'</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span><b>by </b><i><a class="txt" href="jordan6.html#E20:53"><i><font color="Green">E81</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1_1_1_2_1_1"><b>case </b></a><a NAME="E1:53_5_1_1_1_2_1_1"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">Q1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q2'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Q1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q2'</font>;<br/><a NAME="E3:53_5_1_1_1_2_1_1"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E21:53"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:53_5_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E178</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2'</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:53_5_1_1_1_2_1_1"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2'</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E21:53"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:53_5_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1'</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E25:53"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:53_5_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan6.html#E24:53"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:53_5_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E179</font></i>: 
<font color="Maroon">Q1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q1'</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_5_1_1_1_2_1_1_1"/>
<font color="Maroon">Q2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q1'</font>
 ;<br/><a NAME="E2:53_5_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E180</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Q1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q1'</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan6.html#T10" target="_self">Th10</a>, <a class="ref" href="jordan6.html#T11" target="_self">Th11</a>, , <a class="ref" href="jordan6.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan1.html#T7">JORDAN1:7</a></i>;<br/><a NAME="E3:53_5_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">Q1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q1'</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, , <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><b>hence </b><a NAME="E4:53_5_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan6.html#E4:53"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan6.html#E5:53"><i><font color="Green">E52</font></i></a>, , <a class="txt" href="jordan6.html#E1:53"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E10:53_5_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E182</font></i>: 
<font color="Maroon">Q2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1'</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="E11:53_5_1_1_1_2_1_1"/>
<font color="Maroon">Q2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q2'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1_1_1_2_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:53_5_1_1_1_2_1_1_2"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font>
 ;<br/>

<a NAME="E2:53_5_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:53_5_1_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1'</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2'</font> )
 
<b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:53_5_1_1_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2'</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:53_5_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E184</font></i>: 
<font color="Maroon">P2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2'</font>
 <b>by </b><i>, <a class="txt" href="jordan6.html#E14:53"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/><a NAME="E13:53_5_1_1_1_2_1_1"/>
<font color="Maroon">Q1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1_1_1_2_1_1_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:53_5_1_1_1_2_1_1_3"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1'</font>
 ;<br/>

<a NAME="E2:53_5_1_1_1_2_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:53_5_1_1_1_2_1_1_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:53_5_1_1_1_2_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:53_5_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">P1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/><b>hence </b><a NAME="E15:53_5_1_1_1_2_1_1"/>
( ( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font> ) or ( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> ) )
 <b>by </b><i>, , <a class="ref" href="jordan6.html#T4" target="_self">Th4</a>, , , <a class="ref" href="jordan6.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1_1_1_2_1_2"><b>case </b></a><a NAME="E1:53_5_1_1_1_2_1_2"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">Q2'</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Q2'</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q1</font>;<br/><a NAME="E3:53_5_1_1_1_2_1_2"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2'</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E21:53"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:53_5_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E189</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q1</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:53_5_1_1_1_2_1_2"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E21:53"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:53_5_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E25:53"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:53_5_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">Q2'</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan6.html#E24:53"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:53_5_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E190</font></i>: 
<font color="Maroon">Q2'</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_5_1_1_1_2_1_2_1"/>
<font color="Maroon">Q1'</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q2</font>
 ;<br/><a NAME="E2:53_5_1_1_1_2_1_2_1"/><b>then </b><i><font color="Green">E191</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Q2'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q1'</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan6.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan6.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan1.html#T7">JORDAN1:7</a></i>;<br/><a NAME="E3:53_5_1_1_1_2_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">Q2'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="jordan6.html#T9" target="_self">Th9</a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><b>hence </b><a NAME="E4:53_5_1_1_1_2_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan6.html#E4:53"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan6.html#E5:53"><i><font color="Green">E52</font></i></a>, , <a class="txt" href="jordan6.html#E1:53"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E10:53_5_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E192</font></i>: 
<font color="Maroon">Q1'</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q2</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="E11:53_5_1_1_1_2_1_2"/>
<font color="Maroon">Q1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1_1_1_2_1_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:53_5_1_1_1_2_1_2_2"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1'</font>
 ;<br/>

<a NAME="E2:53_5_1_1_1_2_1_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2'</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:53_5_1_1_1_2_1_2_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font> )
 
<b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:53_5_1_1_1_2_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:53_5_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E194</font></i>: 
<font color="Maroon">P1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/><a NAME="E13:53_5_1_1_1_2_1_2"/>
<font color="Maroon">Q2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q2'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1_1_1_2_1_2_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:53_5_1_1_1_2_1_2_3"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font>
 ;<br/>

<a NAME="E2:53_5_1_1_1_2_1_2_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q1</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:53_5_1_1_1_2_1_2_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2'</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1'</font> )
 
<b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:53_5_1_1_1_2_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2'</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:53_5_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">P2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2'</font>
 <b>by </b><i>, <a class="txt" href="jordan6.html#E14:53"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/><b>hence </b><a NAME="E15:53_5_1_1_1_2_1_2"/>
( ( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font> ) or ( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> ) )
 <b>by </b><i>, , <a class="ref" href="jordan6.html#T4" target="_self">Th4</a>, , , <a class="ref" href="jordan6.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:53_5_1_1_1"/>
( ( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font> ) or ( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1_1_2"><b>case </b></a><a NAME="E1:53_5_1_1_2"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">P2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P1'</font>
 ;<br/><div class="add"><div><i><font color="Green">E197</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_5_1_1_2_1"/><i><font color="Green">E198</font></i>: 
( <font color="Maroon">Q2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q1'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q1'</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><a NAME="E2:53_5_1_1_2_1"/><b>then </b><i><font color="Green">E199</font></i>: 
<font color="Maroon">Q2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q1'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:53_5_1_1_2_1"/>
<font color="Maroon">Q1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q2</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E21:53"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>hence </b><a NAME="E4:53_5_1_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="jordan6.html#E17:53"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan6.html#E24:53"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_5_1_1_2_2_1"/>
( <font color="Maroon">Q2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q1'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Q1'</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span><b>by </b><i><a class="txt" href="jordan6.html#E20:53"><i><font color="Green">E81</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:53_5_1_1_2_2_1_1"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">Q2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q1'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Q2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q1'</font>;<br/><a NAME="E3:53_5_1_1_2_2_1_1"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E21:53"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:53_5_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E233</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q1'</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:53_5_1_1_2_2_1_1"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1'</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E21:53"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:53_5_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2'</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E25:53"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:53_5_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">Q2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q2'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan6.html#E24:53"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:53_5_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E234</font></i>: 
<font color="Maroon">Q2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q2'</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1_1_2_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_5_1_1_2_2_1_1_1"/>
<font color="Maroon">Q1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q2'</font>
 ;<br/><a NAME="E2:53_5_1_1_2_2_1_1_1"/><b>then </b><i><font color="Green">E235</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Q2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2'</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q1</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan6.html#T10" target="_self">Th10</a>, <a class="ref" href="jordan6.html#T11" target="_self">Th11</a>, , <a class="ref" href="jordan6.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan1.html#T7">JORDAN1:7</a></i>;<br/><a NAME="E3:53_5_1_1_2_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">Q2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2'</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, , <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><b>hence </b><a NAME="E4:53_5_1_1_2_2_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan6.html#E4:53"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan6.html#E5:53"><i><font color="Green">E52</font></i></a>, , <a class="txt" href="jordan6.html#E1:53"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E10:53_5_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E236</font></i>: 
<font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q2'</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="E11:53_5_1_1_2_2_1_1"/>
<font color="Maroon">Q1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q1'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1_1_2_2_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:53_5_1_1_2_2_1_1_2"/><i><font color="Green">E237</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font>
 ;<br/>

<a NAME="E2:53_5_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q1</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:53_5_1_1_2_2_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2'</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1'</font> )
 
<b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:53_5_1_1_2_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1'</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:53_5_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E238</font></i>: 
<font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1'</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/><a NAME="E13:53_5_1_1_2_2_1_1"/>
<font color="Maroon">Q2'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1_1_2_2_1_1_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:53_5_1_1_2_2_1_1_3"/><i><font color="Green">E239</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2'</font>
 ;<br/>

<a NAME="E2:53_5_1_1_2_2_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q1</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:53_5_1_1_2_2_1_1_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:53_5_1_1_2_2_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:53_5_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">P2'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font>
 <b>by </b><i>, <a class="txt" href="jordan6.html#E14:53"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/><b>hence </b><a NAME="E15:53_5_1_1_2_2_1_1"/>
( ( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font> ) or ( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> ) )
 <b>by </b><i>, , <a class="ref" href="jordan6.html#T4" target="_self">Th4</a>, , , <a class="ref" href="jordan6.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:53_5_1_1_2_2_1_2"/><i><font color="Green">E240</font></i>: 
<font color="Maroon">Q1'</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Q1'</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Q2</font>;<br/><a NAME="E3:53_5_1_1_2_2_1_2"/><i><font color="Green">E241</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1'</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E21:53"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:53_5_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E242</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:53_5_1_1_2_2_1_2"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E21:53"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:53_5_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E25:53"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:53_5_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">Q1'</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan6.html#E24:53"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:53_5_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E243</font></i>: 
<font color="Maroon">Q1'</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1_1_2_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_5_1_1_2_2_1_2_1"/>
<font color="Maroon">Q2'</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q1</font>
 ;<br/><a NAME="E2:53_5_1_1_2_2_1_2_1"/><b>then </b><i><font color="Green">E244</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Q1'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2'</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan6.html#T10" target="_self">Th10</a>, , , <a class="ref" href="jordan6.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan1.html#T7">JORDAN1:7</a></i>;<br/><a NAME="E3:53_5_1_1_2_2_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">Q1'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="jordan6.html#T8" target="_self">Th8</a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><b>hence </b><a NAME="E4:53_5_1_1_2_2_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan6.html#E4:53"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan6.html#E5:53"><i><font color="Green">E52</font></i></a>, , <a class="txt" href="jordan6.html#E1:53"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E10:53_5_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E245</font></i>: 
<font color="Maroon">Q2'</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</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="E11:53_5_1_1_2_2_1_2"/>
<font color="Maroon">Q2'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1_1_2_2_1_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:53_5_1_1_2_2_1_2_2"/><i><font color="Green">E246</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2'</font>
 ;<br/>

<a NAME="E2:53_5_1_1_2_2_1_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q1'</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:53_5_1_1_2_2_1_2_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font> )
 
<b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:53_5_1_1_2_2_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:53_5_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E247</font></i>: 
<font color="Maroon">P2'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font>
 <b>by </b><i>, <a class="txt" href="jordan6.html#E14:53"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/><a NAME="E13:53_5_1_1_2_2_1_2"/>
<font color="Maroon">Q1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q1'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_1_1_2_2_1_2_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:53_5_1_1_2_2_1_2_3"/><i><font color="Green">E248</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font>
 ;<br/>

<a NAME="E2:53_5_1_1_2_2_1_2_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q2'</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:53_5_1_1_2_2_1_2_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1'</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2'</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:53_5_1_1_2_2_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1'</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:53_5_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1'</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/><b>hence </b><a NAME="E15:53_5_1_1_2_2_1_2"/>
( ( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font> ) or ( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> ) )
 <b>by </b><i>, , <a class="ref" href="jordan6.html#T4" target="_self">Th4</a>, , , <a class="ref" href="jordan6.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:53_5_1_1_2"/>
( ( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font> ) or ( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:53_5"/>
( ( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font> ) or ( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> ) )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E68:53"/>
( ( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font> ) or ( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2'</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1'</font> ) )
 ;<br/>


</div>
