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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span><br/> for <font color="Olive" title="b2">p</font> being <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">f</font> holds <br/>( <font color="Olive" title="b1">f</font> is <a href="waybel_0.html#V19" title="WAYBEL_0:attr.19">meet-preserving</a> &amp; <font color="Olive" title="b1">f</font> is <a href="waybel_0.html#V20" title="WAYBEL_0:attr.20">join-preserving</a> )</span><br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">f</font> holds <br/>( <font color="Maroon" title="c2">f</font> is <a href="waybel_0.html#V19" title="WAYBEL_0:attr.19">meet-preserving</a> &amp; <font color="Maroon" title="c2">f</font> is <a href="waybel_0.html#V20" title="WAYBEL_0:attr.20">join-preserving</a> )</span><br/><b>let </b><font color="Maroon" title="c3">p</font> be  <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">f</font> implies ( <font color="Maroon" title="c2">f</font> is <a href="waybel_0.html#V19" title="WAYBEL_0:attr.19">meet-preserving</a> &amp; <font color="Maroon" title="c2">f</font> is <a href="waybel_0.html#V20" title="WAYBEL_0:attr.20">join-preserving</a> ) )</span><br/>





<b>assume </b><a NAME="E1:52"/>
 <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is <a href="waybel_0.html#V19" title="WAYBEL_0:attr.19">meet-preserving</a> &amp; <font color="Maroon" title="c2">f</font> is <a href="waybel_0.html#V20" title="WAYBEL_0:attr.20">join-preserving</a> )</span><br/>

<a NAME="E2:52"/><b>then </b>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  iff <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">p</font> )
 
<b>by </b><i><a class="ref" href="waybel_6.html#T32" target="_self" title="WAYBEL_6:th.32">Th32</a></i>;<br/>
<b>hence </b><a NAME="E3:52"/>
( <font color="Maroon" title="c2">f</font> is <a href="waybel_0.html#V19" title="WAYBEL_0:attr.19">meet-preserving</a> &amp; <font color="Maroon" title="c2">f</font> is <a href="waybel_0.html#V20" title="WAYBEL_0:attr.20">join-preserving</a> )
 <b>by </b><i><a class="ref" href="waybel_6.html#T25" target="_self" title="WAYBEL_6:th.25">Th25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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