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

<b>set </b><font color="Maroon" title="c2">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 implies $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> ) &amp; ( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> $2 implies $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> $2 ) );<br/>
<a NAME="E1:76_1_1"/><i><font color="Green" title="E55">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  ex <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_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="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Olive" title="b1">z</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_1_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:76_1_1_1_1_1"/><i><font color="Green" title="E56">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">  ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Olive" title="b1">z</font>]</span><br/><div class="add"><b>take </b>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>, <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>]</span><br/><b>thus </b><a NAME="E2:76_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>, <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:76_1_1_1_1_1"><i><font color="Green" title="E56">A2</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:76_1_1_1_1_2"/><i><font color="Green" title="E56">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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">  ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Olive" title="b1">z</font>]</span><br/><div class="add"><b>take </b>
<font color="Maroon" title="c3">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</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">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">y</font>]</span><br/><b>thus </b><a NAME="E2:76_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">y</font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:76_1_1_1_1_2"><i><font color="Green" title="E56">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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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="c1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:76_1_1"/><i><font color="Green" title="E56">A4</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c3">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="lattice5.html#E1:76_1_1"><i><font color="Green" title="E55">A1</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">f</font> as   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E5:76_1_1"/><i><font color="Green" title="E57">A5</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="lattice5.html#V2" title="LATTICE5:attr.2">zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <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="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D7" target="_self" title="LATTICE5:def.7">LATTICE5:def 7</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> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span><br/>

<b>thus </b><a NAME="E1:76_1_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:76_1_1"><i><font color="Green" title="E56">A4</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="E6:76_1_1"/><i><font color="Green" title="E58">A6</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <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="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">LATTICE5:def 6</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> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">x'</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">y'</font> = <font color="Maroon" title="c6">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_1_3_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:76_1_1_3_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font></span><br/><div class="add"><b>hence </b><a NAME="E2:76_1_1_3_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:76_1_1_3_1_2"/><i><font color="Green" title="E59">A7</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">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="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font></span><br/><div class="add"><b>hence </b><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> = 
<font color="Maroon" title="c8">y'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c7">x'</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:76_1_1"><i><font color="Green" title="E56">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:76_1_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="lattice5.html#E1:76_1_1_3_1_2"><i><font color="Green" title="E59">A7</font></i></a></i>
;<br/> <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><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:76_1_1"/>
<font color="Maroon" title="c4">f</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">z</font> be    <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="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">LATTICE5:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c8">x'</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c9">y'</font> = <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c10">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="c1">L</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_1_4_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">z</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:76_1_1_4_1_1"/><i><font color="Green" title="E59">A8</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font></span><br/><div class="add"><a NAME="E2:76_1_1_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:76_1_1_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:76_1_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="lattice5.html#E1:76_1_1_4_1_1"><i><font color="Green" title="E59">A8</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:76_1_1_4_1_2"/><i><font color="Green" title="E59">A9</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font></span><br/><div class="add"><b>thus </b><a NAME="E2:76_1_1_4_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_4_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_4_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_1_4_1_2_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_4_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:76_1_1_4_1_2_1_1_1"/><i><font color="Green" title="E60">A10</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font></span><br/><div class="add"><a NAME="E2:76_1_1_4_1_2_1_1_1"/>
<font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T24" title="ORDERS_2:th.24">ORDERS_2:24</a></i>;<br/><a NAME="E3:76_1_1_4_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:76_1_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="lattice5.html#E1:76_1_1_4_1_2"><i><font color="Green" title="E59">A9</font></i></a></i>;<br/><a NAME="E4:76_1_1_4_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>;<br/><a NAME="E5:76_1_1_4_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:76_1_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="lattice5.html#E1:76_1_1_4_1_2_1_1_1"><i><font color="Green" title="E60">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E6:76_1_1_4_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:76_1_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="lattice5.html#E1:76_1_1_4_1_2"><i><font color="Green" title="E59">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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_4_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:76_1_1_4_1_2_1_1_2"/><i><font color="Green" title="E60">A11</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font></span><br/><div class="add"><a NAME="E2:76_1_1_4_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">y'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_4_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_4_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_1_4_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">z</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_4_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:76_1_1_4_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E61">A12</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">y'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font></span><br/><div class="add"><a NAME="E2:76_1_1_4_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">y'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">y'</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T24" title="ORDERS_2:th.24">ORDERS_2:24</a></i>;<br/><a NAME="E3:76_1_1_4_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">y'</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:76_1_1_4_1_2_1_1_2_1_1_1"><i><font color="Green" title="E61">A12</font></i></a>, <a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>;<br/><b>hence </b><a NAME="E4:76_1_1_4_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">y'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:76_1_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="lattice5.html#E1:76_1_1_4_1_2_1_1_2_1_1_1"><i><font color="Green" title="E61">A12</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_4_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:76_1_1_4_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E61">A13</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">y'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font></span><br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">y'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">y'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">y'</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">z'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">y'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">y'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">y'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">y'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">y'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">y'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><a NAME="E3:76_1_1_4_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">y'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">y'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><b>hence </b><a NAME="E4:76_1_1_4_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">y'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:76_1_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="lattice5.html#E1:76_1_1_4_1_2_1_1_2_1_1_2"><i><font color="Green" title="E61">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></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:76_1_1_4_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">x'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">z'</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:76_1_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="lattice5.html#E1:76_1_1_4_1_2_1_1_2"><i><font color="Green" title="E60">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E4:76_1_1_4_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:76_1_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="lattice5.html#E1:76_1_1_4_1_2"><i><font color="Green" title="E59">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></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">f</font> as   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="lattice5.html#E5:76_1_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="lattice5.html#E6:76_1_1"><i><font color="Green" title="E58">A6</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">f</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">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> implies <font color="Maroon" title="c5">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">y</font> ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> implies <font color="Maroon" title="c5">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> ) )</span><br/>

<b>thus </b><a NAME="E9:76_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> implies <font color="Maroon" title="c5">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">y</font> ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> implies <font color="Maroon" title="c5">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> ) )
 <b>by </b><i><a class="txt" href="lattice5.html#E3:76_1_1"><i><font color="Green" title="E56">A4</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>
