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<b>let </b><font color="Maroon" title="c1">TS</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">K</font>, <font color="Maroon" title="c3">L</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>;<br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>







<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c2">K</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">x''</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">TS</font> ;<br/>
<a NAME="E3:53"/><i><font color="Green" title="E33">A2</font></i>: 
not <font color="Maroon" title="c1">TS</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 
<b>by </b><i><a class="txt" href="tops_1.html#E1:53"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E4:53"/>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">L</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c3">L</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E5:53"/><b>then </b><i><font color="Green" title="E34">A3</font></i>: 
( not <font color="Maroon" title="c5">x''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">L</font></span>)</span></span>)</span> &amp; not <font color="Maroon" title="c5">x''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c3">L</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:53"/><b>then </b><i><font color="Green" title="E35">A4</font></i>: 
( not <font color="Maroon" title="c5">x''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">L</font></span>)</span> &amp; not <font color="Maroon" title="c5">x''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">L</font></span>)</span> &amp; not <font color="Maroon" title="c5">x''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c3">L</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">Q1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font><b> such that </b><br/><a NAME="E8:53"/><i><font color="Green" title="E36">A5</font></i>: 
<font color="Maroon" title="c6">Q1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E9:53"/><i><font color="Green" title="E37">A6</font></i>: 
<font color="Maroon" title="c5">x''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q1</font>
 <b>and </b><br/><a NAME="E10:53"/><i><font color="Green" title="E38">A7</font></i>: 
( <font color="Maroon" title="c2">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">L</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">Q1</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">Q1</font> )
 <b>by </b><i><a class="txt" href="tops_1.html#E3:53"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="tops_1.html#T61" target="_self" title="TOPS_1:th.61">Th61</a></i>;<br/>
<a NAME="E11:53"/><i><font color="Green" title="E39">A8</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="tops_1.html#E10:53"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E12:53"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T53" title="XBOOLE_1:th.53">XBOOLE_1:53</a></i>;<br/>
<a NAME="E13:53"/><b>then </b><i><font color="Green" title="E40">A9</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">Q1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">Q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="tops_1.html#E11:53"><i><font color="Green" title="E39">A8</font></i></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E14:53"/>
<font color="Maroon" title="c2">K</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">Q1</font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E1:53"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="tops_1.html#E3:53"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="tops_1.html#E8:53"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="tops_1.html#E9:53"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="tops_1.html#T61" target="_self" title="TOPS_1:th.61">Th61</a></i>;<br/>
<a NAME="E15:53"/><b>then </b><i><font color="Green" title="E41">A10</font></i>: 
<font color="Maroon" title="c2">K</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">Q1</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> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">Q2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font><b> such that </b><br/><a NAME="E17:53"/><i><font color="Green" title="E42">A11</font></i>: 
<font color="Maroon" title="c7">Q2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E18:53"/><i><font color="Green" title="E43">A12</font></i>: 
<font color="Maroon" title="c5">x''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q2</font>
 <b>and </b><br/><a NAME="E19:53"/><i><font color="Green" title="E44">A13</font></i>: 
( <font color="Maroon" title="c2">K</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">L</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">Q2</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">Q2</font> )
 <b>by </b><i><a class="txt" href="tops_1.html#E3:53"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="tops_1.html#E6:53"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="tops_1.html#T61" target="_self" title="TOPS_1:th.61">Th61</a></i>;<br/>
<a NAME="E20:53"/><i><font color="Green" title="E45">A14</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="tops_1.html#E19:53"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E21:53"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T54" title="XBOOLE_1:th.54">XBOOLE_1:54</a></i>;<br/>
<a NAME="E22:53"/><b>then </b><i><font color="Green" title="E46">A15</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Q2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Q2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="tops_1.html#E20:53"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E23:53"/>
<font color="Maroon" title="c2">K</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c7">Q2</font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E1:53"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="tops_1.html#E3:53"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="tops_1.html#E17:53"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="tops_1.html#E18:53"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="tops_1.html#T61" target="_self" title="TOPS_1:th.61">Th61</a></i>;<br/>
<a NAME="E24:53"/><b>then </b><i><font color="Green" title="E47">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Q2</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> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E25:53"/>
not <font color="Maroon" title="c5">x''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c3">L</font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E1:53"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="tops_1.html#E5:53"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">Q3</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font><b> such that </b><br/><a NAME="E27:53"/><i><font color="Green" title="E48">A17</font></i>: 
<font color="Maroon" title="c8">Q3</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E28:53"/><i><font color="Green" title="E49">A18</font></i>: 
<font color="Maroon" title="c5">x''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q3</font>
 <b>and </b><br/><a NAME="E29:53"/><i><font color="Green" title="E50">A19</font></i>: 
( <font color="Maroon" title="c3">L</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">Q3</font> or <font color="Maroon" title="c3">L</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">Q3</font> )
 <b>by </b><i><a class="txt" href="tops_1.html#E3:53"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="tops_1.html#T61" target="_self" title="TOPS_1:th.61">Th61</a></i>;<br/>
<div><i><font color="Green" title="E51">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1"/>
<font color="Maroon" title="c3">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E2:53_1"/><b>then </b>
<font color="Maroon" title="c8">Q3</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E3:53_1"/><b>then </b><i><font color="Green" title="E52">A21</font></i>: 
<font color="Maroon" title="c8">Q3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">L</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T44" title="SUBSET_1:th.44">SUBSET_1:44</a></i>;<br/><a NAME="E4:53_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">Q1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q3</font>
 <b>by </b><i><a class="txt" href="tops_1.html#E13:53"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="tops_1.html#E15:53"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="xboole_1.html#T15" title="XBOOLE_1:th.15">XBOOLE_1:15</a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>;<br/><a NAME="E5:53_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">Q1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="tops_1.html#E3:53_1"><i><font color="Green" title="E52">A21</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><b>hence </b><a NAME="E6:53_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>;<br/></div><b>end;</b></div>
<a NAME="E31:53"/><i><font color="Green" title="E52">A22</font></i>: 
<font color="Maroon" title="c6">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q3</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="tops_1.html#E8:53"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="tops_1.html#E27:53"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="tops_1.html#T38" target="_self" title="TOPS_1:th.38">Th38</a></i>;<br/>
<a NAME="E32:53"/>
<font color="Maroon" title="c5">x''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q3</font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E9:53"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="tops_1.html#E28:53"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E33:53"/><b>then </b>
<font color="Maroon" title="c2">K</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q3</font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E1:53"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="tops_1.html#E3:53"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="tops_1.html#E31:53"><i><font color="Green" title="E52">A22</font></i></a>, <a class="ref" href="tops_1.html#T61" target="_self" title="TOPS_1:th.61">Th61</a></i>;<br/>
<a NAME="E34:53"/><b>then </b><i><font color="Green" title="E53">A23</font></i>: 
<font color="Maroon" title="c8">Q3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">L</font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E29:53"><i><font color="Green" title="E50">A19</font></i></a>, <a class="txt" href="tops_1.html#E30:53"><i><font color="Green" title="E51">A20</font></i></a>, <a class="ref" href="subset_1.html#T44" title="SUBSET_1:th.44">SUBSET_1:44</a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E35:53"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">L</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q3</font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E22:53"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="tops_1.html#E24:53"><i><font color="Green" title="E47">A16</font></i></a>, <a class="ref" href="xboole_1.html#T15" title="XBOOLE_1:th.15">XBOOLE_1:15</a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>;<br/>
<a NAME="E36:53"/><b>then </b>
<font color="Maroon" title="c2">K</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>;<br/>
<a NAME="E37:53"/><b>then </b>
<font color="Maroon" title="c2">K</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q3</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>;<br/>
<a NAME="E38:53"/><b>then </b>
<font color="Maroon" title="c2">K</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="tops_1.html#E34:53"><i><font color="Green" title="E53">A23</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E39:53"/><b>then </b><i><font color="Green" title="E54">A24</font></i>: 
<font color="Maroon" title="c2">K</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q3</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E40:53"/><i><font color="Green" title="E55">A25</font></i>: 
<font color="Maroon" title="c7">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q3</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="tops_1.html#E17:53"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="tops_1.html#E27:53"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="tops_1.html#T38" target="_self" title="TOPS_1:th.38">Th38</a></i>;<br/>
<a NAME="E41:53"/>
<font color="Maroon" title="c5">x''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q3</font>
 
<b>by </b><i><a class="txt" href="tops_1.html#E18:53"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="tops_1.html#E28:53"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E42:53"/>
contradiction
 <b>by </b><i><a class="txt" href="tops_1.html#E1:53"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="tops_1.html#E3:53"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="tops_1.html#E39:53"><i><font color="Green" title="E54">A24</font></i></a>, <a class="txt" href="tops_1.html#E40:53"><i><font color="Green" title="E55">A25</font></i></a>, <a class="ref" href="tops_1.html#T61" target="_self" title="TOPS_1:th.61">Th61</a></i>;<br/>


</div>
