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<span class="kw">let </span><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" 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">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#NV4" title="WAYBEL11:NV.4">inaccessible</a> </span><br/><span class="kw">let </span><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" 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">S</font> is  <a href="waybel11.html#NV4" title="WAYBEL11:NV.4">inaccessible</a> </span><br/><span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel11.html#D1" onmouseover="rs('waybel11/D1')" onmouseout="rh()">WAYBEL11:def 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"> (  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="tarski.html#R2" title="TARSKI: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/>





<span class="kw">assume </span><a NAME="E1:27"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">D</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</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="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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="27_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c4">D9</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">D9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D9</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">E</font> = <font color="Maroon" title="c4">D9</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/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:27_1"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">and </span><br/><a NAME="E6:27_1"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T1" onmouseover="rs('waybel_0/T1')" onmouseout="rh()">WAYBEL_0:1</a></span>;<br/>
<a NAME="E7:27_1"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
 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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a></span>;<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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="27_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" 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="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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27_1_1"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:27_1_1"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
 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" 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="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"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E2:27_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a></span>;<br/>
<a NAME="E4:27_1_1"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E1:27_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:27_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="orders_2.html#T2" onmouseover="rs('orders_2/T2')" onmouseout="rh()">ORDERS_2:2</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="E9:27_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E7:27_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="yellow_0.html#D7" onmouseover="rs('yellow_0/D7')" onmouseout="rh()">YELLOW_0:def 7</a></span>;<br/>
<span class="kw">hence </span><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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E6:27_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E7:27_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="yellow_0.html#D9" onmouseover="rs('yellow_0/D9')" onmouseout="rh()">YELLOW_0:def 9</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>
<span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E14">A1</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>
