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<b>let </b><font color="Maroon">L</font> be  <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V1">with_suprema</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>







<b>assume </b><a NAME="E1:36"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X1</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:36"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E7:36"/><i><font color="Green">E30</font></i>: 
 <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">s</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">Y</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#T14">WAYBEL_0:14</a></i>;<br/>
<a NAME="E8:36"/>
 ex <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> holds <br/><font color="Olive">s</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> ) )
 
<b>by </b><i><a class="ref" href="lattice3.html#D10">LATTICE3:def 10</a></i>;<br/>
<a NAME="E9:36"/><b>then </b>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="lattice3.html#D13">LATTICE3:def 13</a></i>;<br/>
<b>hence </b><a NAME="E10:36"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i/>;<br/>


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