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<div class="add">

<b>thus </b><a NAME="E1:50_1_1"/>
 <a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a><br/>

<b>reconsider </b><font color="Maroon" title="c6">K</font> = <font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c7">L</font> = <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c8">M</font> = <font color="Maroon" title="c5">w</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substlat.html#K1" title="SUBSTLAT:func.1">SubstitutionSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> <b>by </b><i><a class="ref" href="substlat.html#D4" target="_self" title="SUBSTLAT:def.4">Def4</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">u</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">w</font></span>)</span> = 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">K</font>,<span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">L</font>,<font color="Maroon" title="c8">M</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">w</font></span>)</span>
<b>by </b><i><a class="txt" href="substlat.html#E41"><i><font color="Green" title="E35">Lm12</font></i></a></i>
;<br/>


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

<b>thus </b><a NAME="E2:50_1_1"/>
 <a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:50_1_1_2"/>
 <a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D15" title="LATTICES:def.15">LATTICES:def 15</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c3">L</font> =  <a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">E</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substlat.html#K1" title="SUBSTLAT:func.1">SubstitutionSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> <b>by </b><i><a class="ref" href="substlat.html#T1" target="_self" title="SUBSTLAT:th.1">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">e</font> = <font color="Maroon" title="c4">E</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span> <b>by </b><i><a class="ref" href="substlat.html#D4" target="_self" title="SUBSTLAT:def.4">Def4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D13" title="LATTICES:def.13">LATTICES:def 13</a><br/>

<b>let </b><font color="Maroon" title="c6">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">K</font> = <font color="Maroon" title="c6">u</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substlat.html#K1" title="SUBSTLAT:func.1">SubstitutionSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> <b>by </b><i><a class="ref" href="substlat.html#D4" target="_self" title="SUBSTLAT:def.4">Def4</a></i>;<br/>
<font color="Maroon" title="c5">e</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">u</font> = 
 <a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="substlat.html#K2" title="SUBSTLAT:func.2">\/</a> <font color="Maroon" title="c7">K</font></span>)</span>
<b>by </b><i><a class="ref" href="substlat.html#D4" target="_self" title="SUBSTLAT:def.4">Def4</a></i>
<br/>.= 
<font color="Maroon" title="c6">u</font>
<b>by </b><i><a class="ref" href="substlat.html#T11" target="_self" title="SUBSTLAT:th.11">Th11</a></i>
;<br/>
<a NAME="E5:50_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c5">e</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c6">u</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#D9" title="LATTICES:def.9">LATTICES:def 9</a></i>;<br/>
<b>hence </b><a NAME="E6:50_1_1_2_1"/>
( <font color="Maroon" title="c5">e</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c6">u</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> )
 ;<br/>


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

<b>thus </b><a NAME="E2:50_1_1_2"/>
 <a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c3">L</font> =  <a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">E</font> = <span class="p1"><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> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substlat.html#K1" title="SUBSTLAT:func.1">SubstitutionSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> <b>by </b><i><a class="ref" href="substlat.html#T2" target="_self" title="SUBSTLAT:th.2">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">e</font> = <font color="Maroon" title="c4">E</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span> <b>by </b><i><a class="ref" href="substlat.html#D4" target="_self" title="SUBSTLAT:def.4">Def4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14" title="LATTICES:def.14">LATTICES:def 14</a><br/>

<b>let </b><font color="Maroon" title="c6">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">K</font> = <font color="Maroon" title="c6">u</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substlat.html#K1" title="SUBSTLAT:func.1">SubstitutionSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> <b>by </b><i><a class="ref" href="substlat.html#D4" target="_self" title="SUBSTLAT:def.4">Def4</a></i>;<br/>
<font color="Maroon" title="c5">e</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">u</font> = 
 <a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c7">K</font></span>)</span>
<b>by </b><i><a class="ref" href="substlat.html#D4" target="_self" title="SUBSTLAT:def.4">Def4</a></i>
<br/>.= 
 <a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">E</font></span>)</span>
<b>by </b><i><a class="ref" href="substlat.html#T3" target="_self" title="SUBSTLAT:th.3">Th3</a></i>
<br/>.= 
 <a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <font color="Maroon" title="c7">K</font>
<b>by </b><i><a class="ref" href="substlat.html#T4" target="_self" title="SUBSTLAT:th.4">Th4</a></i>
<br/>.= 
<font color="Maroon" title="c6">u</font>
<b>by </b><i><a class="ref" href="substlat.html#T11" target="_self" title="SUBSTLAT:th.11">Th11</a></i>
;<br/>
<a NAME="E5:50_1_1_2_2"/><b>then </b>
( <font color="Maroon" title="c5">e</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c6">u</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#D8" title="LATTICES:def.8">LATTICES:def 8</a></i>;<br/>
<b>hence </b><a NAME="E6:50_1_1_2_2"/>
( <font color="Maroon" title="c5">e</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c6">u</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> )
 ;<br/>


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


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


</div>
