<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>  <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span><br/>  for <font color="Olive" title="b3">X</font>, <font color="Olive" title="b4">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">Y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">X</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Olive" title="b4">Y</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Y</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Olive" title="b2">Y</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c5">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Y</font> implies <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c5">Y</font></span>)</span> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">and </span><br/><a NAME="E2:37"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c5">Y</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">X1</font> = <font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c7">Y1</font> = <font color="Maroon" title="c5">Y</font> as   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">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="yellow_2.html#T41" title="YELLOW_2:th.41">YELLOW_2:41</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">d</font> =  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">X1</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c7">Y1</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T41" title="YELLOW_2:th.41">YELLOW_2:41</a></span>;<br/>
<a NAME="E5:37"/>
<font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_4.html#T29" target="_self" title="YELLOW_4:th.29">Th29</a></span>;<br/>
<a NAME="E6:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A3</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>;<br/>
<a NAME="E7:37"/>
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_4.html#T29" target="_self" title="YELLOW_4:th.29">Th29</a></span>;<br/>
<a NAME="E8:37"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>;<br/>
<a NAME="E9:37"/><span class="kw">then </span>
( <font color="Maroon" title="c8">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c8">d</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></span>;<br/>
<a NAME="E10:37"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T24" title="YELLOW_0:th.24">YELLOW_0:24</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:37"/>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c5">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c5">Y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">Z</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E13:37"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
<font color="Maroon" title="c9">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> or <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font> or  ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b3">b</font> ) ) </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E14:37"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E15:37"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c9">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T44" title="YELLOW_2:th.44">YELLOW_2:44</a></span>;<br/>
<a NAME="E16:37"/>
<font color="Maroon" title="c4">X</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c5">Y</font> or <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c5">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font></span><br/>

<a NAME="E2:37_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">t</font> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:37_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E13:37"><span class="lab"><font color="Green" title="E14">A4</font></span></a></span>; <a class="txt" 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="E17:37"/>
 <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c5">Y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="yellow_3.html#T6" title="YELLOW_3:th.6">YELLOW_3:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
