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<b>let </b><font color="Maroon" title="c1">FT</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c1">FT</font> is <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">symmetric</a>
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c3">C</font> is <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a>
 <b>and </b><br/><a NAME="E3:46"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>and </b><br/><a NAME="E4:46"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c4">P2</font>, <font color="Maroon" title="c5">Q2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="fin_topo.html#D18" title="FIN_TOPO:def.18">FIN_TOPO:def 18</a><br/>

<b>assume </b><a NAME="E5:46"/><i><font color="Green" title="E29">A5</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Q2</font> &amp; <font color="Maroon" title="c4">P2</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>  &amp; <font color="Maroon" title="c5">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>  &amp; <font color="Maroon" title="c4">P2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">Q2</font> )
 ;<br/>

<b>assume </b><a NAME="E6:46"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c4">P2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">Q2</font>
 ;<br/>

<a NAME="E7:46"/>
<font color="Maroon" title="c4">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Q2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E8:46"/><b>then </b><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c4">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> 
 
<b>by </b><i><a class="txt" href="fintopo6.html#E4:46"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="fintopo6.html#E5:46"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">P2</font>;<br/>
<a NAME="E10:46"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P2</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E5:46"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/>
<a NAME="E11:46"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> 
 
<b>by </b><i><a class="txt" href="fintopo6.html#E8:46"><i><font color="Green" title="E31">A7</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font><b> such that </b><br/><a NAME="E13:46"/><i><font color="Green" title="E32">A8</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp;  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c7">z</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )
 ;<br/>
<a NAME="E14:46"/><i><font color="Green" title="E33">A9</font></i>: 
<span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c7">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</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="txt" href="fintopo6.html#E13:46"><i><font color="Green" title="E32">A8</font></i></a>, <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="c8">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c7">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font>;<br/>
<a NAME="E16:46"/><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c7">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E14:46"><i><font color="Green" title="E33">A9</font></i></a></i>;<br/>
<a NAME="E17:46"/><i><font color="Green" title="E35">A11</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c7">z</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> )
 
<b>by </b><i><a class="txt" href="fintopo6.html#E14:46"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">y2</font> = <font color="Maroon" title="c8">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> <b>by </b><i><a class="txt" href="fintopo6.html#E16:46"><i><font color="Green" title="E34">A10</font></i></a></i>;<br/>
<a NAME="E19:46"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c9">y2</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E1:46"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="fintopo6.html#E17:46"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="fin_topo.html#D15" title="FIN_TOPO:def.15">FIN_TOPO:def 15</a></i>;<br/>
<a NAME="E20:46"/><b>then </b>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c9">y2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">P2</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E5:46"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="fintopo6.html#E13:46"><i><font color="Green" title="E32">A8</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="E21:46"/><b>then </b>
 <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c9">y2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">P2</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="E22:46"/><b>then </b><i><font color="Green" title="E36">A12</font></i>: 
<font color="Maroon" title="c9">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> 
 
;<br/>
<a NAME="E23:46"/><i><font color="Green" title="E37">A13</font></i>: 
<font color="Maroon" title="c9">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E14:46"><i><font color="Green" title="E33">A9</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="E24:46"/>
<font color="Maroon" title="c5">Q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">P2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E25:46"/><b>then </b><i><font color="Green" title="E38">A14</font></i>: 
<font color="Maroon" title="c5">Q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> 
 
<b>by </b><i><a class="txt" href="fintopo6.html#E4:46"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="fintopo6.html#E5:46"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">x2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Q2</font>;<br/>
<a NAME="E27:46"/>
<font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q2</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E5:46"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/>
<a NAME="E28:46"/><b>then </b>
<font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> 
 
<b>by </b><i><a class="txt" href="fintopo6.html#E25:46"><i><font color="Green" title="E38">A14</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">z2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font><b> such that </b><br/><a NAME="E30:46"/><i><font color="Green" title="E39">A15</font></i>: 
( <font color="Maroon" title="c11">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font> &amp;  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c11">z2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )
 ;<br/>
<a NAME="E31:46"/><i><font color="Green" title="E40">A16</font></i>: 
<span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c11">z2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</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="txt" href="fintopo6.html#E30:46"><i><font color="Green" title="E39">A15</font></i></a>, <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="c12">y3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c11">z2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font>;<br/>
<a NAME="E33:46"/><i><font color="Green" title="E41">A17</font></i>: 
<font color="Maroon" title="c12">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c11">z2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E31:46"><i><font color="Green" title="E40">A16</font></i></a></i>;<br/>
<a NAME="E34:46"/><i><font color="Green" title="E42">A18</font></i>: 
( <font color="Maroon" title="c12">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c11">z2</font> &amp; <font color="Maroon" title="c12">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> )
 
<b>by </b><i><a class="txt" href="fintopo6.html#E31:46"><i><font color="Green" title="E40">A16</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">y4</font> = <font color="Maroon" title="c12">y3</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> <b>by </b><i><a class="txt" href="fintopo6.html#E33:46"><i><font color="Green" title="E41">A17</font></i></a></i>;<br/>
<a NAME="E36:46"/>
<font color="Maroon" title="c11">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c13">y4</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E1:46"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="fintopo6.html#E34:46"><i><font color="Green" title="E42">A18</font></i></a>, <a class="ref" href="fin_topo.html#D15" title="FIN_TOPO:def.15">FIN_TOPO:def 15</a></i>;<br/>
<a NAME="E37:46"/><b>then </b>
<font color="Maroon" title="c11">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c13">y4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Q2</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E5:46"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="fintopo6.html#E30:46"><i><font color="Green" title="E39">A15</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="E38:46"/><b>then </b><i><font color="Green" title="E43">A19</font></i>: 
 <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c13">y4</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">Q2</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="E39:46"/><i><font color="Green" title="E44">A20</font></i>: 
<font color="Maroon" title="c13">y4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E31:46"><i><font color="Green" title="E40">A16</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">P3</font> = <font color="Maroon" title="c4">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font>;<br/>
<b>set </b><font color="Maroon" title="c15">Q3</font> = <font color="Maroon" title="c5">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font>;<br/>
<i><font color="Green" title="E45">A21</font></i>: <font color="Maroon" title="c3">C</font> = 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E3:46"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</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="c5">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font></span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E5:46"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
;<br/>
<a NAME="E41:46"/>
<font color="Maroon" title="c4">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E42:46"/><b>then </b><i><font color="Green" title="E46">A22</font></i>: 
<font color="Maroon" title="c4">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">Q2</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E5:46"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E43:46"/>
<font color="Maroon" title="c5">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Q2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E44:46"/><b>then </b><i><font color="Green" title="E47">A23</font></i>: 
<font color="Maroon" title="c4">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E42:46"><i><font color="Green" title="E46">A22</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>
<div><i><font color="Green" title="E48">A24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_2"/>
<font color="Maroon" title="c4">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</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="E2:46_2"/><b>then </b>
<font color="Maroon" title="c9">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E23:46"><i><font color="Green" title="E37">A13</font></i></a>, <a class="txt" href="fintopo6.html#E40:46"><i><font color="Green" title="E45">A21</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="E3:46_2"/><b>then </b>
<font color="Maroon" title="c9">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Q2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E22:46"><i><font color="Green" title="E36">A12</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="E4:46_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo6.html#E6:46"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_3"/>
<font color="Maroon" title="c5">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</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="E2:46_3"/><b>then </b><i><font color="Green" title="E49">A25</font></i>: 
<font color="Maroon" title="c13">y4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E39:46"><i><font color="Green" title="E44">A20</font></i></a>, <a class="txt" href="fintopo6.html#E40:46"><i><font color="Green" title="E45">A21</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font><b> such that </b><br/><a NAME="E4:46_3"/><i><font color="Green" title="E50">A26</font></i>: 
( <font color="Maroon" title="c16">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y4</font> &amp;  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c16">w</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">Q2</font> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E38:46"><i><font color="Green" title="E43">A19</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c17">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:46_3"/><i><font color="Green" title="E51">A27</font></i>: 
( <font color="Maroon" title="c17">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c16">w</font> &amp; <font color="Maroon" title="c17">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q2</font> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:46_3"><i><font color="Green" title="E50">A26</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">s2</font> = <font color="Maroon" title="c17">s</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> <b>by </b><i><a class="txt" href="fintopo6.html#E6:46_3"><i><font color="Green" title="E51">A27</font></i></a></i>;<br/><a NAME="E8:46_3"/>
<font color="Maroon" title="c16">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c18">s2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:46"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="fintopo6.html#E6:46_3"><i><font color="Green" title="E51">A27</font></i></a>, <a class="ref" href="fin_topo.html#D15" title="FIN_TOPO:def.15">FIN_TOPO:def 15</a></i>;<br/><a NAME="E9:46_3"/><b>then </b>
 <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c18">s2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">P2</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E2:46_3"><i><font color="Green" title="E49">A25</font></i></a>, <a class="txt" href="fintopo6.html#E4:46_3"><i><font color="Green" title="E50">A26</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E10:46_3"/><b>then </b>
<font color="Maroon" title="c18">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> 
 ;<br/><b>hence </b><a NAME="E11:46_3"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo6.html#E6:46"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="fintopo6.html#E6:46_3"><i><font color="Green" title="E51">A27</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E47:46"/><b>then </b><i><font color="Green" title="E49">A28</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E2:46"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="fintopo6.html#E40:46"><i><font color="Green" title="E45">A21</font></i></a>, <a class="txt" href="fintopo6.html#E44:46"><i><font color="Green" title="E47">A23</font></i></a>, <a class="txt" href="fintopo6.html#E45:46"><i><font color="Green" title="E48">A24</font></i></a>, <a class="ref" href="fin_topo.html#D18" title="FIN_TOPO:def.18">FIN_TOPO:def 18</a></i>;<br/>
<a NAME="E48:46"/>
<font color="Maroon" title="c4">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E49:46"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> 
 
<b>by </b><i><a class="ref" href="fin_topo.html#T19" title="FIN_TOPO:th.19">FIN_TOPO:19</a></i>;<br/>
<a NAME="E50:46"/><b>then </b><i><font color="Green" title="E50">A29</font></i>: 
<font color="Maroon" title="c4">P2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E47:46"><i><font color="Green" title="E49">A28</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E51:46"/>
<font color="Maroon" title="c5">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Q2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E52:46"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo6.html#E6:46"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="fintopo6.html#E50:46"><i><font color="Green" title="E50">A29</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>


</div>
