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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds  <font color="Olive" title="b1">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a></span><br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a></span><br/><b>let </b><font color="Maroon" title="c3">D</font> be  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">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D1" title="WAYBEL11:def.1">WAYBEL11:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> or  not <font color="Maroon" title="c3">D</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">S</font> )</span><br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E14">A1</font></i>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c3">D</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">S</font></span><br/>

<a NAME="E2:27"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c4">D'</font> = <font color="Maroon" title="c3">D</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E2:27_1"/>
<font color="Maroon" title="c4">D'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D'</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">E</font> = <font color="Maroon" title="c4">D'</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">D</font> ;<br/>
<b>consider </b><font color="Maroon" title="c6">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E5:27_1"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 <b>and </b><br/><a NAME="E6:27_1"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c5">E</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T1" title="WAYBEL_0:th.1">WAYBEL_0:1</a></i>;<br/>
<a NAME="E7:27_1"/><i><font color="Green" title="E17">A4</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>  st <font color="Maroon" title="c3">D</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="txt" href="#E5:27_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E8:27_1"/>
for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>  st <font color="Maroon" title="c3">D</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">c</font> &amp; ( for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>  st <font color="Maroon" title="c3">D</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b2">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">c</font> ) holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">D</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c7">c</font> &amp; ( for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>  st <font color="Maroon" title="c3">D</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">c</font> ) implies <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:27_1_1"/><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c3">D</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c7">c</font>
 <b>and </b><br/><a NAME="E2:27_1_1"/><i><font color="Green" title="E19">A6</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>  st <font color="Maroon" title="c3">D</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font></span><br/>

<a NAME="E3:27_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c7">c</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">x</font> )
 
<b>by </b><i><a class="txt" href="#E5:27_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="#E6:27_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E1:27_1_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="#E2:27_1_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:27_1_1"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:27_1"/><b>then </b>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="#E6:27_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E7:27_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="yellow_0.html#D7" title="YELLOW_0:def.7">YELLOW_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E10:27_1"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 <b>by </b><i><a class="txt" href="#E5:27_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="#E6:27_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E7:27_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:27"/>
not <font color="Maroon" title="c3">D</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="#E1:27"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
