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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</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">  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel10.html#K9" title="WAYBEL10:func.9">Subalgebras</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K1" title="WAYBEL10:func.1">opp</a> </span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="waybel10.html#NV1" title="WAYBEL10:NV.1">closure</a> <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a> <a href="waybel10.html#NM1" title="WAYBEL10:NM.1">System</a> of <font color="Maroon" title="c1">L</font>;<br/>
<div><i><font color="Green" title="E34">A1</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" title="c2">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p3">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>] implies <font color="Maroon" title="c2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p3">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>] implies <font color="Maroon" title="c2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p3">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> )</span>
<div class="add"><b>assume </b><a NAME="E1:61_1_1"/>
<font color="Maroon" title="c2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p3">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>]</span><br/><b>then consider </b><font color="Maroon" title="c3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span><b> such that </b><br/><a NAME="E3:61_1_1"/><i><font color="Green" title="E35">A2</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T12" title="YELLOW_2:th.12">YELLOW_2:12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel10.html#T27" target="_self" title="WAYBEL10:th.27">Th27</a></i>;<br/><font color="Maroon" title="c2">a</font> = 
<span class="p1">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="waybel10.html#E3:61_1_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="waybel10.html#T7" target="_self" title="WAYBEL10:th.7">Th7</a></i>
<br/>.= 
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="waybel10.html#D5" target="_self" title="WAYBEL10:def.5">Def5</a></i>
;<br/><b>hence </b><a NAME="E6:61_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c2">a</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">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p3">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span></span><br/><b>then reconsider </b><font color="Maroon" title="c3">S</font> = <font color="Maroon" title="c2">a</font> as  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="waybel10.html#NV1" title="WAYBEL10:NV.1">closure</a> <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a> <a href="waybel10.html#NM1" title="WAYBEL10:NM.1">System</a> of <font color="Maroon" title="c1">L</font> ;<br/><b>reconsider </b><font color="Maroon" title="c4">x</font> =  <a href="waybel10.html#K7" title="WAYBEL10:func.7">closure_op</a> <font color="Maroon" title="c3">S</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span> <b>by </b><i><a class="ref" href="waybel10.html#T27" target="_self" title="WAYBEL10:th.27">Th27</a></i>;<br/><font color="Maroon" title="c3">S</font> = 
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p1">(<span class="default"><a href="waybel10.html#K7" title="WAYBEL10:func.7">closure_op</a> <font color="Maroon" title="c3">S</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel10.html#T19" target="_self" title="WAYBEL10:th.19">Th19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="waybel10.html#K7" title="WAYBEL10:func.7">closure_op</a> <font color="Maroon" title="c3">S</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel10.html#D5" target="_self" title="WAYBEL10:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="waybel10.html#T7" target="_self" title="WAYBEL10:th.7">Th7</a></i>
;<br/><a NAME="E6:61_1"/><b>then </b>
<font color="Maroon" title="c3">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/><b>hence </b><a NAME="E7:61_1"/>
<font color="Maroon" title="c2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p3">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E2:61"/><i><font color="Green" title="E35">A3</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K9" title="WAYBEL10:func.9">Subalgebras</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K1" title="WAYBEL10:func.1">opp</a> </span>)</span> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] )
 
<b>by </b><i><a class="ref" href="waybel10.html#T28" target="_self" title="WAYBEL10:th.28">Th28</a></i>;<br/>
<a NAME="E3:61"/>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p3">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p3">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K9" title="WAYBEL10:func.9">Subalgebras</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K1" title="WAYBEL10:func.1">opp</a> </span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K9" title="WAYBEL10:func.9">Subalgebras</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K1" title="WAYBEL10:func.1">opp</a> </span>)</span> #)
 
<b>from </b><i><a class="ref" href="waybel10.html#S3" target="_self" title="WAYBEL10:sch.3">WAYBEL10:sch 3</a>(<a class="txt" href="waybel10.html#E1:61"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="waybel10.html#E2:61"><i><font color="Green" title="E35">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:61"/>
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel10.html#K6" title="WAYBEL10:func.6">ClImageMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K2" title="WAYBEL10:func.2">|</a> <span class="p2">(<span class="default"><a href="waybel10.html#K8" title="WAYBEL10:func.8">DsupClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel10.html#K9" title="WAYBEL10:func.9">Subalgebras</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="waybel10.html#K1" title="WAYBEL10:func.1">opp</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
