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

<b>let </b><font color="Maroon" title="c1">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">J</font> being <a href="yellow_1.html#V1" title="YELLOW_1:attr.1">RelStr-yielding</a> <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a> <a href="waybel_3.html#V5" title="WAYBEL_3:attr.5">reflexive-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Olive" title="b1">J</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Olive" title="b2">i</font> is   <a href="waybel_0.html#NM6" title="WAYBEL_0:NM.6">Semilattice</a> ) holds <br/>for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Olive" title="b1">J</font></span>)</span><br/> for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b3">g</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b4">i</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">J</font> be  <a href="yellow_1.html#V1" title="YELLOW_1:attr.1">RelStr-yielding</a> <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a> <a href="waybel_3.html#V5" title="WAYBEL_3:attr.5">reflexive-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</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">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Olive" title="b1">i</font> is   <a href="waybel_0.html#NM6" title="WAYBEL_0:NM.6">Semilattice</a> ) implies for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font></span>)</span><br/> for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b3">i</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E13">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Olive" title="b1">i</font> is   <a href="waybel_0.html#NM6" title="WAYBEL_0:NM.6">Semilattice</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font></span>)</span><br/> for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b3">i</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font></span>)</span>; <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">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</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="c3">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span><br/>



<a NAME="E2:30"/><i><font color="Green" title="E14">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Olive" title="b1">i</font> is <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E1:30"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/>
<a NAME="E3:30"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c5">i</font> is   <a href="waybel_0.html#NM6" title="WAYBEL_0:NM.6">Semilattice</a>
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E1:30"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/>
<a NAME="E4:30"/><i><font color="Green" title="E16">A4</font></i>: 
(  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font> is <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> &amp;  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font> is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a> )
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E1:30"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="lfuzzy_0.html#E2:30"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="lfuzzy_0.html#T10" target="_self" title="LFUZZY_0:th.10">Th10</a>, <a class="ref" href="waybel_3.html#T30" title="WAYBEL_3:th.30">WAYBEL_3:30</a></i>;<br/>
<a NAME="E5:30"/><i><font color="Green" title="E17">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_1"/>
( <font color="Maroon" title="c3">f</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c4">g</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">g</font> )
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E4:30"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/>
<a NAME="E2:30_1"/><b>then </b>
( <font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font> )
 
<b>by </b><i><a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></i>;<br/>
<b>hence </b><a NAME="E3:30_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E3:30"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</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="E6:30"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>) <b>-&gt; </b>   <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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> $1</span>)</span> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> $1</span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c6">z</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E2:30_2"/><i><font color="Green" title="E18">A6</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></i>
<a NAME="E3:30_2"/><i><font color="Green" title="E19">A7</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:30_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span><br/>

<a NAME="E1:30_2_1"/>
<font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E2:30_2"><i><font color="Green" title="E18">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:30_2_1"/>
<font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c7">i</font></span>)</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="c7">z</font> = <font color="Maroon" title="c6">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font></span>)</span> <b>by </b><i><a class="txt" href="lfuzzy_0.html#E3:30_2"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="waybel_3.html#T27" title="WAYBEL_3:th.27">WAYBEL_3:27</a></i>;<br/>
<a NAME="E6:30_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Maroon" title="c7">z</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b1">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c7">z</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b1">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b1">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</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">z</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c7">z</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font> )</span><br/>

<a NAME="E1:30_2_2"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c2">J</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c8">i</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> 
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E1:30"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/>
<a NAME="E2:30_2_2"/>
<font color="Maroon" title="c7">z</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E2:30_2"><i><font color="Green" title="E18">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:30_2_2"/>
( <font color="Maroon" title="c7">z</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c7">z</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E1:30_2_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</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="E7:30_2"/><b>then </b>
( <font color="Maroon" title="c7">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c7">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">g</font> )
 
<b>by </b><i><a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></i>;<br/>
<a NAME="E8:30_2"/><b>then </b><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">g</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="lfuzzy_0.html#E4:30"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/>
<a NAME="E9:30_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</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="c3">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><br/>

<a NAME="E1:30_2_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">z</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font> )
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E2:30_2"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="lfuzzy_0.html#E8:30_2"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></i>;<br/>
<b>hence </b><a NAME="E2:30_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</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>hence </b><a NAME="E10:30_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</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>hence </b><a NAME="E7:30"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E3:30"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="lfuzzy_0.html#E5:30"><i><font color="Green" title="E17">A5</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>
