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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="osalg_2.html#K16" target="_self">OSSubAlLattice</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>thus </b><a NAME="E1:82"/>
 <a href="osalg_2.html#K16" target="_self">OSSubAlLattice</a> <font color="Maroon">c<sub>2</sub></font> is <a href="lattices.html#V13">lower-bounded</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D15">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">c<sub>4</sub></font> =  <a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="osalg_2.html#K10" target="_self">GenOSAlg</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="osalg_2.html#D15" target="_self">Def15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K16" target="_self">OSSubAlLattice</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D13">LATTICES:def 13</a><br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K16" target="_self">OSSubAlLattice</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as  <a href="msualg_1.html#V4">strict</a> <a href="osalg_2.html#NM3">OSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="osalg_2.html#D15" target="_self">Def15</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="osalg_2.html#K10" target="_self">GenOSAlg</a> <span class="p2">(<span class="default"><a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_2.html#K11">/\</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="osalg_2.html#D17" target="_self">Def17</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="osalg_2.html#T41" target="_self">Th41</a></i>
;<br/>

<b>hence </b><a NAME="E6:82_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


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

<b>thus </b><a NAME="E2:82"/>
 <a href="osalg_2.html#K16" target="_self">OSSubAlLattice</a> <font color="Maroon">c<sub>2</sub></font> is <a href="lattices.html#V14">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">E56</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="osalg_1.html#T17">OSALG_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="osalg_2.html#E1:82_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="osalg_2.html#D2" target="_self">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="osalg_2.html#K10" target="_self">GenOSAlg</a> <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="osalg_2.html#D15" target="_self">Def15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K16" target="_self">OSSubAlLattice</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14">LATTICES:def 14</a><br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K16" target="_self">OSSubAlLattice</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as  <a href="msualg_1.html#V4">strict</a> <a href="osalg_2.html#NM3">OSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="osalg_2.html#D15" target="_self">Def15</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="osalg_2.html#K10" target="_self">GenOSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="osalg_2.html#K11" target="_self">"\/"_os</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="osalg_2.html#D16" target="_self">Def16</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="osalg_2.html#T43" target="_self">Th43</a></i>
;<br/>

<b>hence </b><a NAME="E9:82_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>


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


</div>
