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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel10.html#K4" title="WAYBEL10:func.4">Sub</a> <font color="Maroon" title="c1">L</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">X</font>, <a href="waybel10.html#K4" title="WAYBEL10:func.4">Sub</a> <font color="Maroon" title="c1">L</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1"><b>now </b></a><div class="add"><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> ex <font color="Olive" title="b1">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> );<br/><b>consider </b><font color="Maroon" title="c3">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:33_1_1_1_1"/><i><font color="Green" title="E18">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i><a NAME="E3:33_1_1_1_1"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )</span><br/>

<b>thus </b><a NAME="E1:33_1_1_1_1_1"/>
(  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="txt" href="#E2:33_1_1_1_1"><i><font color="Green" title="E18">A1</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><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Olive" title="b1">R</font> );<br/><b>consider </b><font color="Maroon" title="c4">S</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:33_1_1_1_1"/><i><font color="Green" title="E20">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i><a NAME="E6:33_1_1_1_1"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> )</span><br/>

<b>thus </b><a NAME="E1:33_1_1_1_1_2"/>
(  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="txt" href="#E5:33_1_1_1_1"><i><font color="Green" title="E20">A3</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="E7:33_1_1_1_1"/>
<font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:33_1_1_1_1_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<b>then consider </b><font color="Maroon" title="c6">R</font> being    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <b> such that </b><br/><a NAME="E3:33_1_1_1_1_3"/><i><font color="Green" title="E22">A5</font></i>: 
( <font color="Maroon" title="c6">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c6">R</font> )
 <b>by </b><i><a class="txt" href="#E5:33_1_1_1_1"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">R</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> )</span><br/>

<a NAME="E1:33_1_1_1_1_3_1"/>
<font color="Maroon" title="c6">R</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="#E3:33_1_1_1_1_3"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="waybel10.html#D3" target="_self" title="WAYBEL10:def.3">Def3</a></i>;<br/>
<a NAME="E2:33_1_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E23">A6</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<b>assume </b><a NAME="E3:33_1_1_1_1_3_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font></span><br/>

<b>hence </b><a NAME="E4:33_1_1_1_1_3_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="txt" href="#E2:33_1_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E3:33_1_1_1_1_3"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="#E2:33_1_1_1_1_3_1"><i><font color="Green" title="E23">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>
<a NAME="E5:33_1_1_1_1_3"/><b>then </b>
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">R</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">R</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="#E3:33_1_1_1_1_3"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E6:33_1_1_1_1_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <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>then reconsider </b><font color="Maroon" title="c5">S</font> = <font color="Maroon" title="c4">S</font> as   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <font color="Maroon" title="c3">Y</font> <b>by </b><i><a class="ref" href="relset_1.html#D1" title="RELSET_1:def.1">RELSET_1:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">A</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c5">S</font> #) as    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="#E3:33_1_1_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="#E6:33_1_1_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c6">A</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#K4" title="WAYBEL10:func.4">Sub</a> <font color="Maroon" title="c1">L</font></span>)</span> <b>by </b><i><a class="ref" href="waybel10.html#D3" target="_self" title="WAYBEL10:def.3">Def3</a></i>;<br/><b>take </b><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c7">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">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c8">a</font> &amp; ( for <font color="Olive" title="b1">b</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#K4" title="WAYBEL10:func.4">Sub</a> <font color="Maroon" title="c1">L</font></span>)</span>  st <font color="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c8">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">b</font> ) )</span><br/><b>thus </b><a NAME="E11:33_1_1_1_1"/>
<font color="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c8">a</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">b</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#K4" title="WAYBEL10:func.4">Sub</a> <font color="Maroon" title="c1">L</font></span>)</span>  st <font color="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c8">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">b</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel10.html#K4" title="WAYBEL10:func.4">Sub</a> <font color="Maroon" title="c1">L</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> or <font color="Maroon" title="c9">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">a</font> )</span><br/>

<b>assume </b><a NAME="E1:33_1_1_1_1_4"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">a</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c10">R</font> = <font color="Maroon" title="c9">b</font> as  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel10.html#D3" target="_self" title="WAYBEL10:def.3">Def3</a></i>;<br/>
<a NAME="E3:33_1_1_1_1_4"/><i><font color="Green" title="E23">A8</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">R</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> )</span><br/>

<b>assume </b><a NAME="E1:33_1_1_1_1_4_1"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font></span><br/>

<a NAME="E2:33_1_1_1_1_4_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<b>hence </b><a NAME="E3:33_1_1_1_1_4_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="txt" href="#E2:33_1_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E1:33_1_1_1_1_4"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="#E1:33_1_1_1_1_4_1"><i><font color="Green" title="E24">A9</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="E4:33_1_1_1_1_4"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c10">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c5">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c10">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font> )</span><br/>

<b>assume </b><a NAME="E1:33_1_1_1_1_4_2"/><i><font color="Green" title="E24">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c10">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font></span><br/>

<a NAME="E2:33_1_1_1_1_4_2"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c10">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<b>hence </b><a NAME="E3:33_1_1_1_1_4_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 <b>by </b><i><a class="txt" href="#E5:33_1_1_1_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="#E1:33_1_1_1_1_4"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="#E1:33_1_1_1_1_4_2"><i><font color="Green" title="E24">A10</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="E5:33_1_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c10">R</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c6">A</font>
 
<b>by </b><i><a class="txt" href="#E3:33_1_1_1_1_4"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<b>hence </b><a NAME="E6:33_1_1_1_1_4"/>
<font color="Maroon" title="c9">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="ref" href="waybel10.html#T16" target="_self" title="WAYBEL10:th.16">Th16</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><b>let </b><font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel10.html#K4" title="WAYBEL10:func.4">Sub</a> <font color="Maroon" title="c1">L</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c9">b</font> implies <font color="Maroon" title="c8">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">b</font> )</span><br/><b>assume </b><a NAME="E12:33_1_1_1_1"/><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c9">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">b</font></span><br/><b>reconsider </b><font color="Maroon" title="c10">B</font> = <font color="Maroon" title="c9">b</font> as  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel10.html#D3" target="_self" title="WAYBEL10:def.3">Def3</a></i>;<br/><a NAME="E14:33_1_1_1_1"/><i><font color="Green" title="E23">A12</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">B</font> )</span><br/>

<b>assume </b><a NAME="E1:33_1_1_1_1_5"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">B</font></span><br/>

<b>then consider </b><font color="Maroon" title="c12">R</font> being    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <b> such that </b><br/><a NAME="E3:33_1_1_1_1_5"/><i><font color="Green" title="E24">A13</font></i>: 
( <font color="Maroon" title="c12">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">R</font> )
 <b>by </b><i><a class="txt" href="#E2:33_1_1_1_1"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">c</font> = <font color="Maroon" title="c12">R</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#K4" title="WAYBEL10:func.4">Sub</a> <font color="Maroon" title="c1">L</font></span>)</span> <b>by </b><i><a class="txt" href="#E3:33_1_1_1_1_5"><i><font color="Green" title="E24">A13</font></i></a></i>;<br/>
<a NAME="E5:33_1_1_1_1_5"/>
<font color="Maroon" title="c13">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="txt" href="#E12:33_1_1_1_1"><i><font color="Green" title="E22">A11</font></i></a>, <a class="txt" href="#E3:33_1_1_1_1_5"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E6:33_1_1_1_1_5"/><b>then </b>
<font color="Maroon" title="c12">R</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c10">B</font>
 
<b>by </b><i><a class="ref" href="waybel10.html#T16" target="_self" title="WAYBEL10:th.16">Th16</a></i>;<br/>
<a NAME="E7:33_1_1_1_1_5"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">B</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<b>hence </b><a NAME="E8:33_1_1_1_1_5"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">B</font>
 <b>by </b><i><a class="txt" href="#E3:33_1_1_1_1_5"><i><font color="Green" title="E24">A13</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="E15:33_1_1_1_1"/>
<font color="Maroon" title="c5">S</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c10">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c10">B</font> )</span><br/>

<b>assume </b><a NAME="E1:33_1_1_1_1_6"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c10">B</font></span><br/>

<b>then consider </b><font color="Maroon" title="c13">R</font> being    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <b> such that </b><br/><a NAME="E3:33_1_1_1_1_6"/><i><font color="Green" title="E24">A14</font></i>: 
( <font color="Maroon" title="c13">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c13">R</font> )
 <b>by </b><i><a class="txt" href="#E5:33_1_1_1_1"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">c</font> = <font color="Maroon" title="c13">R</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#K4" title="WAYBEL10:func.4">Sub</a> <font color="Maroon" title="c1">L</font></span>)</span> <b>by </b><i><a class="txt" href="#E3:33_1_1_1_1_6"><i><font color="Green" title="E24">A14</font></i></a></i>;<br/>
<a NAME="E5:33_1_1_1_1_6"/>
<font color="Maroon" title="c14">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="txt" href="#E12:33_1_1_1_1"><i><font color="Green" title="E22">A11</font></i></a>, <a class="txt" href="#E3:33_1_1_1_1_6"><i><font color="Green" title="E24">A14</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E6:33_1_1_1_1_6"/><b>then </b>
<font color="Maroon" title="c13">R</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c10">B</font>
 
<b>by </b><i><a class="ref" href="waybel10.html#T16" target="_self" title="WAYBEL10:th.16">Th16</a></i>;<br/>
<a NAME="E7:33_1_1_1_1_6"/><b>then </b>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c13">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c10">B</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<b>hence </b><a NAME="E8:33_1_1_1_1_6"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c10">B</font>
 <b>by </b><i><a class="txt" href="#E3:33_1_1_1_1_6"><i><font color="Green" title="E24">A14</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="E16:33_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">a</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c10">B</font>
 <b>by </b><i><a class="txt" href="#E14:33_1_1_1_1"><i><font color="Green" title="E23">A12</font></i></a>, <a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/><b>hence </b><a NAME="E17:33_1_1_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="ref" href="waybel10.html#T16" target="_self" title="WAYBEL10:th.16">Th16</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><b>hence </b><a NAME="E2:33_1_1_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">X</font>, <a href="waybel10.html#K4" title="WAYBEL10:func.4">Sub</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T15" title="YELLOW_0:th.15">YELLOW_0:15</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>
<b>hence </b><a NAME="E2:33_1_1"/>
 <a href="waybel10.html#K4" title="WAYBEL10:func.4">Sub</a> <font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#T53" title="YELLOW_0:th.53">YELLOW_0:53</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>
