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<div class="add">

<b>let </b><font color="Maroon" title="c1">R1</font>, <font color="Maroon" title="c2">R2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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="c1">R1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R1</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="c2">R2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">R2</font> #) &amp; <font color="Maroon" title="c1">R1</font> is   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> implies <font color="Maroon" title="c2">R2</font> is   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:80"/><i><font color="Green" title="E50">A1</font></i>: 
 <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="c1">R1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R1</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="c2">R2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">R2</font> #)
 <b>and </b><br/><a NAME="E2:80"/><i><font color="Green" title="E51">A2</font></i>: 
<font color="Maroon" title="c1">R1</font> is   <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"> <font color="Maroon" title="c2">R2</font> is   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a></span><br/>

<a NAME="E3:80"/><i><font color="Green" title="E52">A3</font></i>: 
<font color="Maroon" title="c2">R2</font> is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D11" title="LATTICE3:def.11">LATTICE3:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">R2</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">R2</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> or <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">y'</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R1</font> <b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">z'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R1</font><b> such that </b><br/><a NAME="E3:80_1"/><i><font color="Green" title="E53">A4</font></i>: 
( <font color="Maroon" title="c7">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x'</font> &amp; <font color="Maroon" title="c7">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y'</font> )
 <b>and </b><br/><a NAME="E4:80_1"/><i><font color="Green" title="E54">A5</font></i>: 
for <font color="Olive" title="b1">w'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R1</font>  st <font color="Olive" title="b1">w'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x'</font> &amp; <font color="Olive" title="b1">w'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y'</font> holds <br/><font color="Olive" title="b1">w'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">z'</font>
 <b>by </b><i><a class="txt" href="#E2:80"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="lattice3.html#D11" title="LATTICE3:def.11">LATTICE3:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c7">z'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R2</font> <b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">z</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">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c8">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">R2</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">z</font> ) ) )</span><br/>

<b>thus </b><a NAME="E6:80_1"/>
( <font color="Maroon" title="c8">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c8">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> )
 <b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="#E3:80_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">R2</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">z</font> )</span><br/>

<b>let </b><font color="Maroon" title="c9">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R2</font>; <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">w</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> or  not <font color="Maroon" title="c9">w</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> or <font color="Maroon" title="c9">w</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">z</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c10">w'</font> = <font color="Maroon" title="c9">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R1</font> <b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E8:80_1"/>
( <font color="Maroon" title="c9">w</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c9">w</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">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="c9">w</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">z</font></span><br/>

<a NAME="E9:80_1"/><b>then </b>
( <font color="Maroon" title="c10">w'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x'</font> &amp; <font color="Maroon" title="c10">w'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y'</font> )
 
<b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E10:80_1"/><b>then </b>
<font color="Maroon" title="c10">w'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">z'</font>
 
<b>by </b><i><a class="txt" href="#E4:80_1"><i><font color="Green" title="E54">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:80_1"/>
<font color="Maroon" title="c9">w</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">z</font>
 <b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green" title="E1">Lm1</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="E4:80"/><i><font color="Green" title="E53">A6</font></i>: 
<font color="Maroon" title="c2">R2</font> is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D10" title="LATTICE3:def.10">LATTICE3:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">R2</font> st <br/>( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">R2</font>  holds <br/>(  not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">y'</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R1</font> <b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">z'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R1</font><b> such that </b><br/><a NAME="E3:80_2"/><i><font color="Green" title="E54">A7</font></i>: 
( <font color="Maroon" title="c7">z'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">x'</font> &amp; <font color="Maroon" title="c7">z'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">y'</font> )
 <b>and </b><br/><a NAME="E4:80_2"/><i><font color="Green" title="E55">A8</font></i>: 
for <font color="Olive" title="b1">w'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R1</font>  st <font color="Olive" title="b1">w'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">x'</font> &amp; <font color="Olive" title="b1">w'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">y'</font> holds <br/><font color="Olive" title="b1">w'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">z'</font>
 <b>by </b><i><a class="txt" href="#E2:80"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="lattice3.html#D10" title="LATTICE3:def.10">LATTICE3:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c7">z'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R2</font> <b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">z</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">R2</font>  holds <br/>(  not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c8">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/>

<b>thus </b><a NAME="E6:80_2"/>
( <font color="Maroon" title="c8">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c8">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">y</font> )
 <b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="#E3:80_2"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">R2</font>  holds <br/>(  not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c8">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<b>let </b><font color="Maroon" title="c9">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">w</font> or  not <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">w</font> or <font color="Maroon" title="c8">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">w</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c10">w'</font> = <font color="Maroon" title="c9">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R1</font> <b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E8:80_2"/>
( <font color="Maroon" title="c9">w</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c9">w</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">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="c8">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">w</font></span><br/>

<a NAME="E9:80_2"/><b>then </b>
( <font color="Maroon" title="c10">w'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">x'</font> &amp; <font color="Maroon" title="c10">w'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">y'</font> )
 
<b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E10:80_2"/><b>then </b>
<font color="Maroon" title="c10">w'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">z'</font>
 
<b>by </b><i><a class="txt" href="#E4:80_2"><i><font color="Green" title="E55">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:80_2"/>
<font color="Maroon" title="c9">w</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">z</font>
 <b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green" title="E1">Lm1</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="E5:80"/><i><font color="Green" title="E54">A9</font></i>: 
<font color="Maroon" title="c2">R2</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D1" title="YELLOW_0:def.1">YELLOW_0:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c4">x'</font> = <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R1</font> <b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a></i>;<br/>
<a NAME="E2:80_3"/>
<font color="Maroon" title="c4">x'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">x'</font>
 
<b>by </b><i><a class="txt" href="#E2:80"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="yellow_0.html#D1" title="YELLOW_0:def.1">YELLOW_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:80_3"/>
<font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green" title="E1">Lm1</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="E6:80"/><i><font color="Green" title="E55">A10</font></i>: 
<font color="Maroon" title="c2">R2</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</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="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> or  not <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">z</font> or <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">z</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c6">x'</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c7">y'</font> = <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c8">z'</font> = <font color="Maroon" title="c5">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R1</font> <b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:80_4"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">z</font></span><br/>

<a NAME="E3:80_4"/><b>then </b>
( <font color="Maroon" title="c6">x'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y'</font> &amp; <font color="Maroon" title="c7">y'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">z'</font> )
 
<b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E4:80_4"/><b>then </b>
<font color="Maroon" title="c6">x'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">z'</font>
 
<b>by </b><i><a class="txt" href="#E2:80"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:80_4"/>
<font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green" title="E1">Lm1</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="E7:80"/>
<font color="Maroon" title="c2">R2</font> is <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D3" title="YELLOW_0:def.3">YELLOW_0: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="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> or  not <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">y'</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R1</font> <b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:80_5"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">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="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font></span><br/>

<a NAME="E3:80_5"/><b>then </b>
( <font color="Maroon" title="c5">x'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y'</font> &amp; <font color="Maroon" title="c6">y'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x'</font> )
 
<b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:80_5"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="#E2:80"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="yellow_0.html#D3" title="YELLOW_0:def.3">YELLOW_0:def 3</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="E8:80"/>
<font color="Maroon" title="c2">R2</font> is   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 <b>by </b><i><a class="txt" href="#E3:80"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="#E4:80"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="#E5:80"><i><font color="Green" title="E54">A9</font></i></a>, <a class="txt" href="#E6:80"><i><font color="Green" title="E55">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>
