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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font>  st <font color="Maroon" title="c1">FT</font> is  <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">symmetric</a>  &amp; <font color="Olive" title="b2">C</font> is  <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a>  &amp; <font color="Olive" title="b2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">C</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  holds <br/><font color="Olive" title="b1">A</font> is  <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">FT</font> is  <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">symmetric</a>  &amp; <font color="Maroon" title="c3">C</font> is  <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a>  &amp; <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> &amp; <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>  implies <font color="Maroon" title="c2">A</font> is  <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:46"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c1">FT</font> is  <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">symmetric</a> 
 <span class="kw">and </span><br/><a NAME="E2:46"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c3">C</font> is  <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a> 
 <span class="kw">and </span><br/><a NAME="E3:46"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:46"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A</font> is  <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a> </span><br/>

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="fin_topo.html#D18" title="FIN_TOPO:def.18">FIN_TOPO:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or <font color="Maroon" title="c4">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c5">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  not <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> or  not <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> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:46"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:46"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E7:46"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E8:46"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <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></span><br/>

<span class="kw">assume </span><a NAME="E9:46"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">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="E11:46"/><span class="lab"><font color="Green" title="E34">A10</font></span>: 
<font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46"><span class="lab"><font color="Green" title="E31">A7</font></span></a></span>;<br/>
<a NAME="E12:46"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E13:46"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E14:46"/><span class="kw">then </span>
<font color="Maroon" title="c6">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:46"><span class="lab"><font color="Green" title="E34">A10</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">z2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font><span class="kw"> such that </span><br/><a NAME="E16:46"/><span class="lab"><font color="Green" title="E35">A11</font></span>: 
<font color="Maroon" title="c7">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font>
 <span class="kw">and </span><br/><a NAME="E17:46"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
 <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c7">z2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">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="c7">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="E19:46"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
<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">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:46"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E20:46"/><span class="kw">then </span>
<font color="Maroon" title="c8">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="c7">z2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">y4</font> = <font color="Maroon" title="c8">y3</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> ;<br/>
<a NAME="E22:46"/>
<font color="Maroon" title="c8">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="c7">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:46"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E23:46"/><span class="kw">then </span>
<font color="Maroon" title="c7">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="c9">y4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="fin_topo.html#D15" title="FIN_TOPO:def.15">FIN_TOPO:def 15</a></span>;<br/>
<a NAME="E24:46"/><span class="kw">then </span>
<font color="Maroon" title="c7">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="c9">y4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E16:46"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E25:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A14</font></span>: 
 <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c9">y4</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">Q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">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/>
<span class="kw">set </span><font color="Maroon" title="c11">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/>
<span class="lab"><font color="Green" title="E39">A15</font></span>: <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">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="E28:46"/><span class="lab"><font color="Green" title="E40">A16</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E30">A6</font></span></a></span>;<br/>
<a NAME="E29:46"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E30:46"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E31:46"/><span class="kw">then </span>
<font color="Maroon" title="c12">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:46"><span class="lab"><font color="Green" title="E40">A16</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font><span class="kw"> such that </span><br/><a NAME="E33:46"/><span class="lab"><font color="Green" title="E41">A17</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">and </span><br/><a NAME="E34:46"/><span class="lab"><font color="Green" title="E42">A18</font></span>: 
 <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c13">z</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">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="c13">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="E36:46"/><span class="lab"><font color="Green" title="E43">A19</font></span>: 
<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">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:46"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E37:46"/><span class="kw">then </span>
<font color="Maroon" title="c14">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="c13">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">y2</font> = <font color="Maroon" title="c14">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT</font> ;<br/>
<a NAME="E39:46"/>
<font color="Maroon" title="c14">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="c13">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:46"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E40:46"/><span class="kw">then </span>
<font color="Maroon" title="c13">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="c15">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="fin_topo.html#D15" title="FIN_TOPO:def.15">FIN_TOPO:def 15</a></span>;<br/>
<a NAME="E41:46"/><span class="kw">then </span>
<font color="Maroon" title="c13">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="c15">y2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E33:46"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E42:46"/><span class="kw">then </span>
 <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c15">y2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E43:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A20</font></span>: 
<font color="Maroon" title="c15">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="E44:46"/><span class="lab"><font color="Green" title="E45">A21</font></span>: 
<font color="Maroon" title="c9">y4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:46"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E46">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:46_2"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:46_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A23</font></span>: 
<font color="Maroon" title="c9">y4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:46"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E26:46"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:46_2"/><span class="lab"><font color="Green" title="E48">A24</font></span>: 
<font color="Maroon" title="c16">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y4</font>
 <span class="kw">and </span><br/><a NAME="E5:46_2"/><span class="lab"><font color="Green" title="E49">A25</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:46"><span class="lab"><font color="Green" title="E38">A14</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:46_2"/><span class="lab"><font color="Green" title="E50">A26</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:46_2"/><span class="lab"><font color="Green" title="E51">A27</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_2"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_2"><span class="lab"><font color="Green" title="E50">A26</font></span></a></span>;<br/><a NAME="E10:46_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E7:46_2"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="ref" href="fin_topo.html#D15" title="FIN_TOPO:def.15">FIN_TOPO:def 15</a></span>;<br/><a NAME="E11:46_2"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_2"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E4:46_2"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E12:46_2"/><span class="kw">then </span>
<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/><span class="kw">hence </span><a NAME="E13:46_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E8:46_2"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E46:46"/><span class="lab"><font color="Green" title="E47">A28</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E47:46"/><span class="lab"><font color="Green" title="E48">A29</font></span>: 
<font color="Maroon" title="c15">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:46"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E49">A30</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:46_3"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:46_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:46"><span class="lab"><font color="Green" title="E48">A29</font></span></a>, <a class="txt" href="#E26:46"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:46_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:46"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E4:46_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E49:46"/>
<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="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">Q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:46"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/>
<a NAME="E50:46"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/>
<a NAME="E51:46"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E26:46"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="txt" href="#E48:46"><span class="lab"><font color="Green" title="E49">A30</font></span></a>, <a class="txt" href="#E45:46"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="ref" href="fin_topo.html#D18" title="FIN_TOPO:def.18">FIN_TOPO:def 18</a></span>;<br/>
<a NAME="E52:46"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:46"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="ref" href="fin_topo.html#T19" title="FIN_TOPO:th.19">FIN_TOPO:19</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/>
<span class="kw">hence </span><a NAME="E53:46"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
