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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">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="E1:44"/><span class="lab"><font color="Green" title="E20">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="E2:44"/><span class="lab"><font color="Green" title="E21">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="E3:44"/><span class="lab"><font color="Green" title="E22">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="E4:44"/><span class="lab"><font color="Green" title="E23">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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="firebrick">:: according to </font></span><a class="ref" href="fin_topo.html#D16" onmouseover="rs('fin_topo/D16')" onmouseout="rh()">FIN_TOPO:def 16</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="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="E5:44"/><span class="lab"><font color="Green" title="E24">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="E6:44"/><span class="lab"><font color="Green" title="E25">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="E7:44"/><span class="lab"><font color="Green" title="E26">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="E8:44"/><span class="lab"><font color="Green" title="E27">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="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="E9:44"/><span class="lab"><font color="Green" title="E28">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">x2</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Q2</font>;<br/>
<a NAME="E10:44"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:44"><span class="lab"><font color="Green" title="E26">A7</font></span></a></span>;<br/>
<a NAME="E11:44"/>
<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" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E12:44"/><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="#E4:44"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E5:44"><span class="lab"><font color="Green" title="E24">A5</font></span></a></span>;<br/>
<a NAME="E13:44"/><span class="kw">then </span>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Q2</font> <a href="tarski.html#R2" title="TARSKI: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="#E10:44"><span class="lab"><font color="Green" title="E29">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="E15:44"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon" title="c7">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Q2</font>
 <span class="kw">and </span><br/><a NAME="E16:44"/><span class="lab"><font color="Green" title="E31">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">set </span><font color="Maroon" title="c8">y3</font> =  the    <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="E17:44"/><span class="lab"><font color="Green" title="E32">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="#E16:44"><span class="lab"><font color="Green" title="E31">A12</font></span></a></span>;<br/>
<a NAME="E18:44"/><span class="kw">then </span>
 the    <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> <a href="tarski.html#R2" title="TARSKI: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> =  the    <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> 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="E20:44"/>
 the    <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> <a href="tarski.html#R2" title="TARSKI: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="#E17:44"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E21:44"/><span class="kw">then </span>
<font color="Maroon" title="c7">z2</font> <a href="tarski.html#R2" title="TARSKI: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="#E1:44"><span class="lab"><font color="Green" title="E20">A1</font></span></a></span>;<br/>
<a NAME="E22:44"/><span class="kw">then </span>
<font color="Maroon" title="c7">z2</font> <a href="tarski.html#R2" title="TARSKI: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="#E7:44"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E15:44"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E23:44"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">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>
 
;<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="E34">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="#E3:44"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">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="#E5:44"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="xboole_1.html#T23" onmouseover="rs('xboole_1/T23')" onmouseout="rh()">XBOOLE_1:23</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">P2</font>;<br/>
<a NAME="E25:44"/><span class="lab"><font color="Green" title="E35">A16</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">P2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:44"><span class="lab"><font color="Green" title="E25">A6</font></span></a></span>;<br/>
<a NAME="E26:44"/>
<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" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E27:44"/><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="#E4:44"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E5:44"><span class="lab"><font color="Green" title="E24">A5</font></span></a></span>;<br/>
<a NAME="E28:44"/><span class="kw">then </span>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">P2</font> <a href="tarski.html#R2" title="TARSKI: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="#E25:44"><span class="lab"><font color="Green" title="E35">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="E30:44"/><span class="lab"><font color="Green" title="E36">A17</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">P2</font>
 <span class="kw">and </span><br/><a NAME="E31:44"/><span class="lab"><font color="Green" title="E37">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">set </span><font color="Maroon" title="c14">y</font> =  the    <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="E32:44"/><span class="lab"><font color="Green" title="E38">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="#E31:44"><span class="lab"><font color="Green" title="E37">A18</font></span></a></span>;<br/>
<a NAME="E33:44"/><span class="kw">then </span>
 the    <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> <a href="tarski.html#R2" title="TARSKI: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> =  the    <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> 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="E35:44"/>
 the    <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> <a href="tarski.html#R2" title="TARSKI: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="#E32:44"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E36:44"/><span class="kw">then </span>
<font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI: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="#E1:44"><span class="lab"><font color="Green" title="E20">A1</font></span></a></span>;<br/>
<a NAME="E37:44"/><span class="kw">then </span>
<font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI: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="#E6:44"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E30:44"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E38:44"/><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>
 
;<br/>
<a NAME="E39:44"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A20</font></span>: 
<font color="Maroon" title="c15">y2</font> <a href="tarski.html#R2" title="TARSKI: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="E40:44"/><span class="lab"><font color="Green" title="E40">A21</font></span>: 
<font color="Maroon" title="c9">y4</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:44"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E41">A22</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="44_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="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> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:44_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:44_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A23</font></span>: 
<font color="Maroon" title="c9">y4</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:44"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="txt" href="#E24:44"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">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="E4:44_2"/><span class="lab"><font color="Green" title="E43">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="E5:44_2"/><span class="lab"><font color="Green" title="E44">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="#E23:44"><span class="lab"><font color="Green" title="E33">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">object</a> <span class="kw"> such that </span><br/><a NAME="E7:44_2"/><span class="lab"><font color="Green" title="E45">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="E8:44_2"/><span class="lab"><font color="Green" title="E46">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="#E5:44_2"><span class="lab"><font color="Green" title="E44">A25</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">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="#E7:44_2"><span class="lab"><font color="Green" title="E45">A26</font></span></a></span>;<br/><a NAME="E10:44_2"/>
<font color="Maroon" title="c16">w</font> <a href="tarski.html#R2" title="TARSKI: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="#E1:44"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E7:44_2"><span class="lab"><font color="Green" title="E45">A26</font></span></a></span>;<br/><a NAME="E11:44_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="#E2:44_2"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="txt" href="#E4:44_2"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/><a NAME="E12:44_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">s2</font> <a href="tarski.html#R2" title="TARSKI: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="E13:44_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:44"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E8:44_2"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E42:44"/><span class="lab"><font color="Green" title="E42">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" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E43:44"/><span class="lab"><font color="Green" title="E43">A29</font></span>: 
<font color="Maroon" title="c15">y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:44"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E44">A30</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="44_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="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> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:44_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:44_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:44"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E24:44"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:44_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">y2</font> <a href="tarski.html#R2" title="TARSKI: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="#E39:44"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E4:44_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:44"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E45:44"/>
<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="#E8:44"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a>, <a class="ref" href="xboole_1.html#T63" onmouseover="rs('xboole_1/T63')" onmouseout="rh()">XBOOLE_1:63</a></span>;<br/>
<a NAME="E46:44"/><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" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a>, <a class="ref" href="xboole_1.html#T63" onmouseover="rs('xboole_1/T63')" onmouseout="rh()">XBOOLE_1:63</a></span>;<br/>
<a NAME="E47:44"/><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="#E2:44"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E24:44"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="txt" href="#E44:44"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E41:44"><span class="lab"><font color="Green" title="E41">A22</font></span></a></span>;<br/>
<a NAME="E48:44"/><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="#E42:44"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="ref" href="fin_topo.html#T14" onmouseover="rs('fin_topo/T14')" onmouseout="rh()">FIN_TOPO:14</a>, <a class="ref" href="xboole_1.html#T63" onmouseover="rs('xboole_1/T63')" onmouseout="rh()">XBOOLE_1:63</a></span>;<br/>
<span class="kw">hence </span><a NAME="E49:44"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:44"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a>, <a class="ref" href="xboole_1.html#T63" onmouseover="rs('xboole_1/T63')" onmouseout="rh()">XBOOLE_1:63</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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