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<span class="kw">let </span><font color="Maroon" title="c1">N</font> be   <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</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">X</font>, <font color="Olive" title="b2">Y</font> being  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">Y</font> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">Y</font></span>)</span> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font> be   <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> </span><br/>



<span class="kw">thus </span><a NAME="E1:46"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel30.html#T27" target="_self" title="WAYBEL30:th.27">Th27</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> </span>)</span></span><br/>

<span class="kw">assume </span><a NAME="E2:46"/>
not <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> </span>)</span>
 ; <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">then </span><span class="kw">consider </span><font color="Maroon" title="c4">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:46"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> 
 <span class="kw">and </span><br/><a NAME="E5:46"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
not <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E6:46"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
not <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E7:46"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
not <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> <a href="waybel30.html#K1" title="WAYBEL30:func.1">^0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c4">s</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E25">A1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">D</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font><span class="kw"> such that </span><br/><a NAME="E10:46"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<font color="Maroon" title="c5">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">D</font>
 <span class="kw">and </span><br/><a NAME="E11:46"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E27">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">E</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font><span class="kw"> such that </span><br/><a NAME="E13:46"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<font color="Maroon" title="c5">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">E</font>
 <span class="kw">and </span><br/><a NAME="E14:46"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
<font color="Maroon" title="c3">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46"><span class="lab"><font color="Green" title="E28">A4</font></span></a></span>;<br/>
<a NAME="E15:46"/>
<font color="Maroon" title="c5">s</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T25" title="YELLOW_0:th.25">YELLOW_0:25</a></span>;<br/>
<a NAME="E16:46"/><span class="kw">then </span>
<font color="Maroon" title="c5">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">E</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E13:46"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="yellow_3.html#T2" title="YELLOW_3:th.2">YELLOW_3:2</a></span>;<br/>
<a NAME="E17:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A9</font></span>: 
<font color="Maroon" title="c5">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c7">E</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_2.html#T51" title="WAYBEL_2:th.51">WAYBEL_2:51</a></span>;<br/>
<a NAME="E18:46"/>
 ex <font color="Olive" title="b1">xy</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> st <br/>( <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">xy</font> &amp; ( for <font color="Olive" title="b2">D</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>  st <font color="Olive" title="b1">xy</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b2">D</font> holds <br/><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Y</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">D</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E25">A1</font></span></a></span>;<br/>
<a NAME="E19:46"/><span class="kw">then </span>
<font color="Maroon" title="c2">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Y</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">D</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c7">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:46"><span class="lab"><font color="Green" title="E33">A9</font></span></a></span>;<br/>
<a NAME="E20:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c7">E</font></span>)</span> <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="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E21:46"/>
<font color="Maroon" title="c2">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">D</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c7">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:46"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="yellow12.html#T21" title="YELLOW12:th.21">YELLOW12:21</a></span>;<br/>
<a NAME="E22:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A11</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c7">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<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="E23:46"/>
<font color="Maroon" title="c3">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">D</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c7">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:46"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="yellow12.html#T21" title="YELLOW12:th.21">YELLOW12:21</a></span>;<br/>
<a NAME="E24:46"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c7">E</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c7">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:46"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:46"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:46"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</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>
