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<span class="kw">let </span><font color="Maroon" title="c1">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span><br/>  for <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c1">P</font> is  <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a>  &amp; <font color="Olive" title="b2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">p2</font> &amp; <font color="Olive" title="b1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/> not <font color="Olive" title="b1">Q</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c1">P</font> is  <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a>  &amp; <font color="Olive" title="b1">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b2">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/> not <font color="Maroon" title="c2">Q</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">P</font> is  <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a>  &amp; <font color="Maroon" title="c3">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c3">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font> &amp; <font color="Maroon" title="c2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> implies  not <font color="Maroon" title="c2">Q</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c1">P</font> is  <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a> 
 <span class="kw">and </span><br/><a NAME="E2:50"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c3">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">and </span><br/><a NAME="E3:50"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<font color="Maroon" title="c4">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">and </span><br/><a NAME="E4:50"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font>
 <span class="kw">and </span><br/><a NAME="E5:50"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
<font color="Maroon" title="c2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c2">Q</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">P1</font>, <font color="Maroon" title="c6">P2</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E7:50"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<font color="Maroon" title="c5">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 <span class="kw">and </span><br/><a NAME="E8:50"/><span class="lab"><font color="Green" title="E38">A7</font></span>: 
<font color="Maroon" title="c6">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 <span class="kw">and </span><br/><a NAME="E9:50"/><span class="lab"><font color="Green" title="E39">A8</font></span>: 
<font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P2</font>
 <span class="kw">and </span><br/><a NAME="E10:50"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
<font color="Maroon" title="c5">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="topreal2.html#T5" onmouseover="rs('topreal2/T5')" onmouseout="rh()">TOPREAL2:5</a></span>;<br/>
<a NAME="E11:50"/><span class="lab"><font color="Green" title="E41">A10</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">P</font> = <font color="Maroon" title="c1">P</font> as   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E32">A1</font></span></a></span>;<br/>
<a NAME="E13:50"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
<font color="Maroon" title="c5">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E14:50"/>
<font color="Maroon" title="c5">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" onmouseover="rs('xboole_1/T36')" onmouseout="rh()">XBOOLE_1:36</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">P19</font> = <font color="Maroon" title="c5">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">P</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:50"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E13:50"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E16:50"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
<font color="Maroon" title="c6">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E17:50"/>
<font color="Maroon" title="c6">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" onmouseover="rs('xboole_1/T36')" onmouseout="rh()">XBOOLE_1:36</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">P29</font> = <font color="Maroon" title="c6">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">P</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:50"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E16:50"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E19:50"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
<font color="Maroon" title="c8">P19</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="txt" href="#E9:50"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E10:50"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="jordan6.html#T39" target="_self" onmouseover="rs('jordan6/T39')" onmouseout="rh()">Th39</a></span>;<br/>
<a NAME="E20:50"/><span class="lab"><font color="Green" title="E45">A14</font></span>: 
<font color="Maroon" title="c9">P29</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E9:50"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E10:50"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="jordan6.html#T39" target="_self" onmouseover="rs('jordan6/T39')" onmouseout="rh()">Th39</a></span>;<br/>
<a NAME="E21:50"/><span class="lab"><font color="Green" title="E46">A15</font></span>: 
<font color="Maroon" title="c2">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">P19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">P29</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">P29</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_1"/><span class="lab"><font color="Green" title="E47">A16</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">P29</font></span><br/>

<a NAME="E2:50_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A17</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P1</font> &amp;  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">P29</font> ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P2</font> &amp;  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">P29</font> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:50_1_1_1"/>
( ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P1</font> &amp;  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P2</font> &amp;  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E9:50"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:50_1_1_1_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P1</font> &amp;  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">P29</font></span><br/><div class="add"><a NAME="E2:50_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:50_1_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">P29</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:50_1_1_1_2"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P2</font> &amp;  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">P29</font></span><br/><div class="add"><a NAME="E2:50_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P29</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:50_1_1_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">P29</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:50_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">P29</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c10">p3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E23:50"/><span class="lab"><font color="Green" title="E47">A18</font></span>: 
<font color="Maroon" title="c10">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">P1</font>
 <span class="kw">and </span><br/><a NAME="E24:50"/><span class="lab"><font color="Green" title="E48">A19</font></span>: 
<font color="Maroon" title="c10">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p1</font>
 <span class="kw">and </span><br/><a NAME="E25:50"/><span class="lab"><font color="Green" title="E49">A20</font></span>: 
<font color="Maroon" title="c10">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="ref" href="jordan6.html#T42" target="_self" onmouseover="rs('jordan6/T42')" onmouseout="rh()">Th42</a></span>;<br/>
<a NAME="E26:50"/>
 not <font color="Maroon" title="c10">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:50"><span class="lab"><font color="Green" title="E48">A19</font></span></a>, <a class="txt" href="#E25:50"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E27:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A21</font></span>: 
<font color="Maroon" title="c8">P19</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:50"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E28:50"/>
<font color="Maroon" title="c8">P19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Q</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P19</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_2"/><span class="lab"><font color="Green" title="E51">A22</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P19</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font></span><br/>

<a NAME="E2:50_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A23</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E3:50_2"/><span class="lab"><font color="Green" title="E53">A24</font></span>: 
 not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2"><span class="lab"><font color="Green" title="E51">A22</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E4:50_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E2:50_2"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:50_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E3:50_2"><span class="lab"><font color="Green" title="E53">A24</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E29:50"/><span class="kw">then </span>
<font color="Maroon" title="c8">P19</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:50"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/>
<a NAME="E30:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A25</font></span>: 
<font color="Maroon" title="c8">P19</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">Q</font>
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">p39</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E32:50"/><span class="lab"><font color="Green" title="E52">A26</font></span>: 
<font color="Maroon" title="c11">p39</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P2</font>
 <span class="kw">and </span><br/><a NAME="E33:50"/><span class="lab"><font color="Green" title="E53">A27</font></span>: 
<font color="Maroon" title="c11">p39</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p1</font>
 <span class="kw">and </span><br/><a NAME="E34:50"/><span class="lab"><font color="Green" title="E54">A28</font></span>: 
<font color="Maroon" title="c11">p39</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="ref" href="jordan6.html#T42" target="_self" onmouseover="rs('jordan6/T42')" onmouseout="rh()">Th42</a></span>;<br/>
<a NAME="E35:50"/>
 not <font color="Maroon" title="c11">p39</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:50"><span class="lab"><font color="Green" title="E53">A27</font></span></a>, <a class="txt" href="#E34:50"><span class="lab"><font color="Green" title="E54">A28</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E36:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A29</font></span>: 
<font color="Maroon" title="c9">P29</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:50"><span class="lab"><font color="Green" title="E52">A26</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E37:50"/>
<font color="Maroon" title="c9">P29</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Q</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P29</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_3"/><span class="lab"><font color="Green" title="E56">A30</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P29</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font></span><br/>

<a NAME="E2:50_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A31</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E3:50_3"/><span class="lab"><font color="Green" title="E58">A32</font></span>: 
 not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_3"><span class="lab"><font color="Green" title="E56">A30</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E4:50_3"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_3"><span class="lab"><font color="Green" title="E57">A31</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:50_3"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E9:50"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E3:50_3"><span class="lab"><font color="Green" title="E58">A32</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E38:50"/><span class="kw">then </span>
<font color="Maroon" title="c9">P29</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:50"><span class="lab"><font color="Green" title="E55">A29</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/>
<a NAME="E39:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A33</font></span>: 
<font color="Maroon" title="c9">P29</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">Q</font>
 
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c8">P19</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">P29</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:50_4"/>
<font color="Maroon" title="c8">P19</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">P29</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">p0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:50_4"/><span class="lab"><font color="Green" title="E57">A34</font></span>: 
<font color="Maroon" title="c12">p0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P19</font>
 <span class="kw">and </span><br/><a NAME="E4:50_4"/><span class="lab"><font color="Green" title="E58">A35</font></span>: 
<font color="Maroon" title="c12">p0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P29</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/><a NAME="E5:50_4"/><span class="lab"><font color="Green" title="E59">A36</font></span>: 
<font color="Maroon" title="c12">p0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_4"><span class="lab"><font color="Green" title="E57">A34</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E6:50_4"/><span class="lab"><font color="Green" title="E60">A37</font></span>: 
 not <font color="Maroon" title="c12">p0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_4"><span class="lab"><font color="Green" title="E57">A34</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E7:50_4"/>
<font color="Maroon" title="c12">p0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_4"><span class="lab"><font color="Green" title="E58">A35</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E8:50_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:50"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E5:50_4"><span class="lab"><font color="Green" title="E59">A36</font></span></a>, <a class="txt" href="#E6:50_4"><span class="lab"><font color="Green" title="E60">A37</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E41:50"/>
 not <font color="Maroon" title="c2">Q</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:50"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E20:50"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E21:50"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E30:50"><span class="lab"><font color="Green" title="E51">A25</font></span></a>, <a class="txt" href="#E39:50"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="ref" href="topreal5.html#T1" onmouseover="rs('topreal5/T1')" onmouseout="rh()">TOPREAL5:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
