<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c2">S1</font>, <font color="Maroon" title="c3">S2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="waybel10.html#K3" title="WAYBEL10:func.3">ClOpers</a> <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">f</font> being <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> holds <br/> ( <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S1</font> iff <font color="Olive" title="b1">f</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> ) ) &amp; ( for <font color="Olive" title="b1">f</font> being <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> holds <br/> ( <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S2</font> iff <font color="Olive" title="b1">f</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> ) ) implies <font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:57_1_2"/><i><font color="Green" title="E29">A6</font></i>: 
for <font color="Olive" title="b1">f</font> being <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> holds <br/> ( <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S1</font> iff <font color="Olive" title="b1">f</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> )
 <b>and </b><br/><a NAME="E2:57_1_2"/><i><font color="Green" title="E30">A7</font></i>: 
for <font color="Olive" title="b1">f</font> being <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> holds <br/> ( <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S2</font> iff <font color="Olive" title="b1">f</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</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">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font></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="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>;<br/>
<div><i><font color="Green" title="E31">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">f</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="c4">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S1</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">f</font>] )</span><br/><a NAME="E1:57_1_2_1"/>
( <font color="Maroon" title="c4">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S1</font> implies <font color="Maroon" title="c4">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel10.html#K3" title="WAYBEL10:func.3">ClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/><a NAME="E2:57_1_2_1"/><b>then </b>
( <font color="Maroon" title="c4">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S1</font> implies <font color="Maroon" title="c4">f</font> is  <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#T11" target="_self" title="WAYBEL10:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E3:57_1_2_1"/>
( <font color="Maroon" title="c4">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S1</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">f</font>] )
 <b>by </b><i><a class="txt" href="#E1:57_1_2"><i><font color="Green" title="E29">A6</font></i></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>
<div><i><font color="Green" title="E32">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">f</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="c4">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S2</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">f</font>] )</span><br/><a NAME="E1:57_1_2_2"/>
( <font color="Maroon" title="c4">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S2</font> implies <font color="Maroon" title="c4">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel10.html#K3" title="WAYBEL10:func.3">ClOpers</a> <font color="Maroon" title="c1">L</font></span>)</span> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/><a NAME="E2:57_1_2_2"/><b>then </b>
( <font color="Maroon" title="c4">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S2</font> implies <font color="Maroon" title="c4">f</font> is  <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#T11" target="_self" title="WAYBEL10:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E3:57_1_2_2"/>
( <font color="Maroon" title="c4">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S2</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">f</font>] )
 <b>by </b><i><a class="txt" href="#E2:57_1_2"><i><font color="Green" title="E30">A7</font></i></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="E5:57_1_2"/>
 <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 <font color="Maroon" title="c2">S1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S1</font> #) <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 <font color="Maroon" title="c3">S2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">S2</font> #)
 
<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="#E3:57_1_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="#E4:57_1_2"><i><font color="Green" title="E32">A9</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:57_1_2"/>
<font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
