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<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> is  <a href="orders_2.html#V1">strict</a> <a href="yellow_0.html#V7">closure</a> <a href="waybel_0.html#V4">directed-sups-inheriting</a> <a href="waybel10.html#NM1">System</a> of <font color="Maroon">L</font>;<br/>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:61_1_1"/>
<font color="Maroon">a</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">L</font></span>)</span> <a href="waybel10.html#K2" target="_self">|</a> <span class="p3">(<span class="default"><a href="waybel10.html#K8" target="_self">DsupClOpers</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel10.html#K8" target="_self">DsupClOpers</a> <font color="Maroon">L</font></span>)</span><b> such that </b><br/><a NAME="E3:61_1_1"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">L</font></span>)</span> <a href="waybel10.html#K2" target="_self">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" target="_self">DsupClOpers</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T12">YELLOW_2:12</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="waybel_1.html#V7">closure</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<font color="Maroon">L</font> <b>by </b><i/>;<br/><font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E6:61_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>]
 ;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:61_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>]
 ;<br/><b>then reconsider </b><font color="Maroon">S</font> = <font color="Maroon">a</font> as  <a href="orders_2.html#V1">strict</a> <a href="yellow_0.html#V7">closure</a> <a href="waybel_0.html#V4">directed-sups-inheriting</a> <a href="waybel10.html#NM1">System</a> of <font color="Maroon">L</font> ;<br/><b>reconsider </b><font color="Maroon">x</font> =  <a href="waybel10.html#K7" target="_self">closure_op</a> <font color="Maroon">S</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel10.html#K8" target="_self">DsupClOpers</a> <font color="Maroon">L</font></span>)</span> <b>by </b><i/>;<br/><font color="Maroon">S</font> = 
 <a href="yellow_2.html#K1">Image</a> <span class="p1">(<span class="default"><a href="waybel10.html#K7" target="_self">closure_op</a> <font color="Maroon">S</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="waybel10.html#K7" target="_self">closure_op</a> <font color="Maroon">S</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">L</font></span>)</span> <a href="waybel10.html#K2" target="_self">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" target="_self">DsupClOpers</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/><a NAME="E6:61_1"/><b>then </b>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">L</font></span>)</span> <a href="waybel10.html#K2" target="_self">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" target="_self">DsupClOpers</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/><b>hence </b><a NAME="E7:61_1"/>
<font color="Maroon">a</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">L</font></span>)</span> <a href="waybel10.html#K2" target="_self">|</a> <span class="p3">(<span class="default"><a href="waybel10.html#K8" target="_self">DsupClOpers</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:61"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K9" target="_self">Subalgebras</a> <font color="Maroon">L</font></span>)</span> <a href="waybel10.html#K1" target="_self">opp</a> </span>)</span> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>] )
 
<b>by </b><i/>;<br/>
<a NAME="E3:61"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">L</font></span>)</span> <a href="waybel10.html#K2" target="_self">|</a> <span class="p3">(<span class="default"><a href="waybel10.html#K8" target="_self">DsupClOpers</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">L</font></span>)</span> <a href="waybel10.html#K2" target="_self">|</a> <span class="p3">(<span class="default"><a href="waybel10.html#K8" target="_self">DsupClOpers</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K9" target="_self">Subalgebras</a> <font color="Maroon">L</font></span>)</span> <a href="waybel10.html#K1" target="_self">opp</a> </span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K9" target="_self">Subalgebras</a> <font color="Maroon">L</font></span>)</span> <a href="waybel10.html#K1" target="_self">opp</a> </span>)</span> #)
 
<b>from </b><i><a class="ref" href="waybel10.html#S3" target="_self">WAYBEL10:sch 3</a>(<a class="txt" href="waybel10.html#E2:61"><i><font color="Green">E34</font></i></a>, );<br/></i>
<b>hence </b><a NAME="E4:61"/>
 <a href="yellow_2.html#K1">Image</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K6" target="_self">ClImageMap</a> <font color="Maroon">L</font></span>)</span> <a href="waybel10.html#K2" target="_self">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" target="_self">DsupClOpers</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel10.html#K9" target="_self">Subalgebras</a> <font color="Maroon">L</font></span>)</span> <a href="waybel10.html#K1" target="_self">opp</a> 
 ;<br/>


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