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

<b>reconsider </b><font color="Maroon">f</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finsop_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E2:31_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="lattice5.html#NM3">BiFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">L</font> ;<br/>
<a NAME="E4:31_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">f</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:31_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">f</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">y</font>,<font color="Olive">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">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<b>thus </b><font color="Maroon">f</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:31_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> is <a href="lattice5.html#V1" target="_self">symmetric</a>
 
<b>by </b><i/>;<br/>
<a NAME="E7:31_1_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">f</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:31_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> is <a href="lattice5.html#V2" target="_self">zeroed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E9:31_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">y</font>,<font color="Olive">z</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">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">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E1:31_1_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">f</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">f</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:31_1_1_2"/>
<font color="Maroon">f</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:31_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:31_1_1"/><b>then </b>
<font color="Maroon">f</font> is <a href="lattice5.html#V3" target="_self">u.t.i.</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:31_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="lattice5.html#NM3">BiFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="lattice5.html#V1" target="_self">symmetric</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="lattice5.html#V2" target="_self">zeroed</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="lattice5.html#V3" target="_self">u.t.i.</a> )
 <b>by </b><i>, </i>;<br/>


</div>
