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<a NAME="E1:30_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">P</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<b>thus </b><a NAME="E3:30_1_1"/>
not <font color="Maroon">P</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>thus </b><a NAME="E4:30_1_1"/>
<font color="Maroon">P</font> is <a href="knaster.html#V2" target="_self">with_suprema</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="knaster.html#D8" target="_self">KNASTER:def 8</a><br/>

<b>assume </b><a NAME="E1:30_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 ;<br/>

<b>take </b><font color="Maroon">z</font> = <font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font>;<br/>

<b>thus </b><a NAME="E2:30_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>thus </b><a NAME="E3:30_1_1_1"/>
( <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="lattices.html#T22">LATTICES:22</a></i>;<br/>

<b>let </b><font color="Maroon">z'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E4:30_1_1_1"/>
( <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z'</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z'</font> )
 ;<br/>

<b>hence </b><a NAME="E5:30_1_1_1"/>
<font color="Maroon">z</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z'</font>
 <b>by </b><i><a class="ref" href="boolealg.html#T11">BOOLEALG:11</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="knaster.html#D9" target="_self">KNASTER:def 9</a><br/>

<b>assume </b><a NAME="E5:30_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 ;<br/>

<b>take </b><font color="Maroon">z</font> = <font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font>;<br/>

<b>thus </b><a NAME="E6:30_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>thus </b><a NAME="E7:30_1_1"/>
( <font color="Maroon">z</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="lattices.html#T23">LATTICES:23</a></i>;<br/>

<b>let </b><font color="Maroon">z'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E8:30_1_1"/>
( <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">z'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y</font> )
 ;<br/>

<b>hence </b><a NAME="E9:30_1_1"/>
<font color="Maroon">z'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="boolealg.html#T12">BOOLEALG:12</a></i>;<br/>


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