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<b>let </b><font color="Maroon" title="c1">S1</font> be   <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon" title="c2">U0</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font>;<br/>



<b>set </b><font color="Maroon" title="c3">L</font> =  <a href="osalg_2.html#K16" title="OSALG_2:func.16">OSSubAlLattice</a> <font color="Maroon" title="c2">U0</font>;<br/>
<b>thus </b><a NAME="E1:82"/>
 <a href="osalg_2.html#K16" title="OSALG_2:func.16">OSSubAlLattice</a> <font color="Maroon" title="c2">U0</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="82_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c4">C</font> =  <a href="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Maroon" title="c2">U0</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">G</font> =  <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Maroon" title="c2">U0</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="osalg_2.html#K13" title="OSALG_2:func.13">OSSub</a> <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="osalg_2.html#D15" target="_self" title="OSALG_2:def.15">Def15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">G1</font> = <font color="Maroon" title="c5">G</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K16" title="OSALG_2:func.16">OSSubAlLattice</a> <font color="Maroon" title="c2">U0</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c6">G1</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="c7">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K16" title="OSALG_2:func.16">OSSubAlLattice</a> <font color="Maroon" title="c2">U0</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c8">a1</font> = <font color="Maroon" title="c7">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="osalg_2.html#K13" title="OSALG_2:func.13">OSSub</a> <font color="Maroon" title="c2">U0</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">a2</font> = <font color="Maroon" title="c8">a1</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="osalg_2.html#D15" target="_self" title="OSALG_2:def.15">Def15</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c6">G1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">a</font> = 
<span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <span class="p2">(<span class="default"><a href="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Maroon" title="c2">U0</font></span>)</span></span>)</span> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <font color="Maroon" title="c9">a2</font>
<b>by </b><i><a class="ref" href="osalg_2.html#D17" target="_self" title="OSALG_2:def.17">Def17</a></i>
<br/>.= 
<font color="Maroon" title="c6">G1</font>
<b>by </b><i><a class="ref" href="osalg_2.html#T41" target="_self" title="OSALG_2:th.41">Th41</a></i>
;<br/>

<b>hence </b><a NAME="E6:82_1"/>
<font color="Maroon" title="c7">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font>
 ;<br/>


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

<b>thus </b><a NAME="E2:82"/>
 <a href="osalg_2.html#K16" title="OSALG_2:func.16">OSSubAlLattice</a> <font color="Maroon" title="c2">U0</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2"><b>proof </b></a><div class="add">

<a NAME="E1:82_2"/><i><font color="Green" title="E56">A1</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U0</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font>
 
<b>by </b><i><a class="ref" href="osalg_1.html#T17" title="OSALG_1:th.17">OSALG_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">B</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U0</font> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">B</font> = <font color="Maroon" title="c4">B</font> as    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="txt" href="osalg_2.html#E1:82_2"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="osalg_2.html#D2" target="_self" title="OSALG_2:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">G</font> =  <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c5">B</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="osalg_2.html#K13" title="OSALG_2:func.13">OSSub</a> <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="osalg_2.html#D15" target="_self" title="OSALG_2:def.15">Def15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">G1</font> = <font color="Maroon" title="c6">G</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K16" title="OSALG_2:func.16">OSSubAlLattice</a> <font color="Maroon" title="c2">U0</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c7">G1</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="c8">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K16" title="OSALG_2:func.16">OSSubAlLattice</a> <font color="Maroon" title="c2">U0</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c9">a1</font> = <font color="Maroon" title="c8">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="osalg_2.html#K13" title="OSALG_2:func.13">OSSub</a> <font color="Maroon" title="c2">U0</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">a2</font> = <font color="Maroon" title="c9">a1</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="osalg_2.html#D15" target="_self" title="OSALG_2:def.15">Def15</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c7">G1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">a</font> = 
<span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <font color="Maroon" title="c10">a2</font>
<b>by </b><i><a class="ref" href="osalg_2.html#D16" target="_self" title="OSALG_2:def.16">Def16</a></i>
<br/>.= 
<font color="Maroon" title="c7">G1</font>
<b>by </b><i><a class="ref" href="osalg_2.html#T43" target="_self" title="OSALG_2:th.43">Th43</a></i>
;<br/>

<b>hence </b><a NAME="E9:82_2"/>
<font color="Maroon" title="c8">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c7">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G1</font>
 ;<br/>


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


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