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<b>let </b><font color="Maroon" title="c1">S</font> be   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:108"/><i><font color="Green" title="E66">A1</font></i>: 
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>  st  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">S</font> holds <br/>for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Olive" title="b2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">y</font></span>)</span> where <font color="Olive" title="b3">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> </span>}</span> ,<font color="Maroon" title="c1">S</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D3" target="_self" title="WAYBEL_1:def.3">WAYBEL_1:def 3</a><br/>

<b>set </b><font color="Maroon" title="c5">Y</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">s</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:108_1_1"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">s</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c7">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E3:108_1_1"/><i><font color="Green" title="E67">A2</font></i>: 
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c7">s</font>
 <b>and </b><br/><a NAME="E4:108_1_1"/><i><font color="Green" title="E68">A3</font></i>: 
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ;<br/><b>thus </b><a NAME="E5:108_1_1"/>
( <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> )
 <b>by </b><i><a class="txt" href="waybel_1.html#E3:108_1_1"><i><font color="Green" title="E67">A2</font></i></a>, <a class="txt" href="waybel_1.html#E4:108_1_1"><i><font color="Green" title="E68">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:108_1"/><i><font color="Green" title="E67">A4</font></i>: 
( <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_2"/>
( <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> )
 </span><b>by </b><i><a class="txt" href="waybel_1.html#E2:108_1"><i><font color="Green" title="E67">A4</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_1"><b>suppose </b></a><a NAME="E1:108_1_2_1"/><i><font color="Green" title="E68">A5</font></i>: 
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">y</font>
 ;<br/><div class="add"><a NAME="E2:108_1_2_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:108_1_2_1"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">s</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:108_1_2_1"><i><font color="Green" title="E68">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2"><b>suppose </b></a><a NAME="E1:108_1_2_2"/><i><font color="Green" title="E68">A6</font></i>: 
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font>
 ;<br/><div class="add"><a NAME="E2:108_1_2_2"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:108_1_2_2"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">s</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:108_1_2_2"><i><font color="Green" title="E68">A6</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:108"/><b>then </b><i><font color="Green" title="E67">A7</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">s</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E4:108"/><i><font color="Green" title="E68">A8</font></i>: 
 <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="c3">y</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T20" title="YELLOW_0:th.20">YELLOW_0:20</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">z</font></span>)</span> = 
<font color="Maroon" title="c2">x</font> <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> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T41" title="YELLOW_0:th.41">YELLOW_0:41</a></i>
<br/>.= 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">S</font>
<b>by </b><i><a class="txt" href="waybel_1.html#E1:108"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="waybel_1.html#E3:108"><i><font color="Green" title="E67">A7</font></i></a>, <a class="txt" href="waybel_1.html#E4:108"><i><font color="Green" title="E68">A8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T41" title="YELLOW_0:th.41">YELLOW_0:41</a></i>
;<br/>


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