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

<b>reconsider </b><font color="Maroon" title="c3">f</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> as   <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"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</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="c2">L</font> ;<br/>
<a NAME="E2:31_1_1"/><i><font color="Green" title="E20">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <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="c2">L</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<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 <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font> ;<br/>
<a NAME="E4:31_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Maroon" title="c4">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="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><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 <font color="Maroon" title="c1">A</font>;<br/>

<b>thus </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> = 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font>
<b>by </b><i><a class="txt" href="lattice5.html#E2:31_1_1"><i><font color="Green" title="E20">A1</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#E2:31_1_1"><i><font color="Green" title="E20">A1</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:31_1_1"/><b>then </b><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>
 
<b>by </b><i><a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></i>;<br/>
<a NAME="E6:31_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Maroon" title="c4">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">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="c2">L</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E2:31_1_1"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<a NAME="E7:31_1_1"/><b>then </b><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="lattice5.html#V2" title="LATTICE5:attr.2">zeroed</a>
 
<b>by </b><i><a class="ref" href="lattice5.html#D7" target="_self" title="LATTICE5:def.7">Def7</a></i>;<br/>
<a NAME="E8:31_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds  <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="Olive" title="b1">x</font>,<font color="Olive" title="b2">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="Olive" title="b2">y</font>,<font color="Olive" title="b3">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="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2"><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 <font color="Maroon" title="c1">A</font>;<br/>

<a NAME="E1:31_1_1_2"/><i><font color="Green" title="E23">A4</font></i>: 
( <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>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> &amp; <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> <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="c2">L</font> )
 
<b>by </b><i><a class="txt" href="lattice5.html#E2:31_1_1"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<a NAME="E2:31_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="c7">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E2:31_1_1"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:31_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#E1:31_1_1_2"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:31_1_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>
 
<b>by </b><i><a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E10:31_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="lattice5.html#V2" title="LATTICE5:attr.2">zeroed</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> )
 <b>by </b><i><a class="txt" href="lattice5.html#E5:31_1_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="lattice5.html#E7:31_1_1"><i><font color="Green" title="E22">A3</font></i></a></i>;<br/>


</div>
