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

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="lattice5.html#NM3">BiFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">L</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:41"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">d</font> is <a href="lattice5.html#V1" target="_self">symmetric</a>
 <b>and </b><br/><a NAME="E2:41"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">d</font> is <a href="lattice5.html#V3" target="_self">u.t.i.</a>
 ;<br/>

<b>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K4">:]</a></span>;<br/>

<b>assume </b><a NAME="E3:41"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K10">`3</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K11">`4</a> </span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">x</font> = <font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> ;<br/>
<b>set </b><font color="Maroon">y</font> = <font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> ;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">q</font> <a href="mcart_1.html#K10">`3</a> , <font color="Maroon">b</font> = <font color="Maroon">q</font> <a href="mcart_1.html#K11">`4</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E6:41"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:41_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font>, <font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/>
<a NAME="E3:41_1"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">u'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:41_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:41_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:41_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:41"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:41_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_2_1"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) )
 </span><b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_1"><b>suppose </b></a><a NAME="E1:41_2_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/><a NAME="E3:41_2_1_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_2_1_1"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:41_2_1_1"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 <b>by </b><i/>;<br/><a NAME="E6:41_2_1_1"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E7:41_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><b>hence </b><a NAME="E8:41_2_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_2"><b>suppose </b></a><a NAME="E1:41_2_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/><a NAME="E3:41_2_1_2"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_2_1_2"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:41_2_1_2"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 <b>by </b><i/>;<br/><a NAME="E6:41_2_1_2"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E7:41_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><a NAME="E8:41_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><a NAME="E9:41_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/><b>hence </b><a NAME="E10:41_2_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_3"><b>suppose </b></a><a NAME="E1:41_2_1_3"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/><a NAME="E3:41_2_1_3"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_2_1_3"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:41_2_1_3"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 <b>by </b><i/>;<br/><a NAME="E6:41_2_1_3"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E7:41_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><b>hence </b><a NAME="E8:41_2_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E8:41"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:41_3"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_3_1"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) )
 </span><b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_1_1"><b>suppose </b></a><a NAME="E1:41_3_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/><a NAME="E3:41_3_1_1"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_3_1_1"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:41_3_1_1"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><i><font color="Green">E73</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><a NAME="E7:41_3_1_1"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">u'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<font color="Maroon">u'</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E8:41_3_1_1"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">u'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E9:41_3_1_1"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">u'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E10:41_3_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_1_2"><b>suppose </b></a><a NAME="E1:41_3_1_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/><a NAME="E3:41_3_1_2"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_3_1_2"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:41_3_1_2"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><i><font color="Green">E78</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><a NAME="E7:41_3_1_2"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">u'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<font color="Maroon">u'</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E8:41_3_1_2"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">u'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E9:41_3_1_2"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">u'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E10:41_3_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_1_3"><b>suppose </b></a><a NAME="E1:41_3_1_3"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/><a NAME="E3:41_3_1_3"/><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_3_1_3"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:41_3_1_3"/><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><i><font color="Green">E87</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><a NAME="E7:41_3_1_3"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">u'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<font color="Maroon">u'</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E8:41_3_1_3"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">u'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E9:41_3_1_3"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<font color="Maroon">u'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E10:41_3_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E9:41"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:41_4"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_4_1"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) )
 </span><b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_1"><b>suppose </b></a><a NAME="E1:41_4_1_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:41_4_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_2"><b>suppose </b></a><a NAME="E1:41_4_1_2"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E3:41_4_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_3"><b>suppose </b></a><a NAME="E1:41_4_1_3"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_4_1_3"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><a NAME="E3:41_4_1_3"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_4_1_3"/><b>then </b><i><font color="Green">E96</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><a NAME="E5:41_4_1_3"/><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/><a NAME="E6:41_4_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E7:41_4_1_3"/><b>then </b><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:41"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow_3.html#T3">YELLOW_3:3</a></i>;<br/><a NAME="E8:41_4_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, </i>;<br/><a NAME="E9:41_4_1_3"/><b>then </b><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E11:41_4_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_4"><b>suppose </b></a><a NAME="E1:41_4_1_4"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i>, </i>
;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
;<br/><b>hence </b><a NAME="E4:41_4_1_4"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_5"><b>suppose </b></a><a NAME="E1:41_4_1_5"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:41_4_1_5"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_6"><b>suppose </b></a><a NAME="E1:41_4_1_6"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_4_1_6"/><b>then </b><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E3:41_4_1_6"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_4_1_6"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><a NAME="E5:41_4_1_6"/><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/><a NAME="E6:41_4_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E7:41_4_1_6"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:41"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow_3.html#T3">YELLOW_3:3</a></i>;<br/><a NAME="E8:41_4_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, </i>;<br/><a NAME="E9:41_4_1_6"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E11:41_4_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_7"><b>suppose </b></a><a NAME="E1:41_4_1_7"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_4_1_7"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E3:41_4_1_7"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_4_1_7"/><b>then </b><i><font color="Green">E113</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><a NAME="E5:41_4_1_7"/><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/><a NAME="E6:41_4_1_7"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E1:41"><i><font color="Green">E29</font></i></a>, </i>;<br/><a NAME="E7:41_4_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E8:41_4_1_7"/><b>then </b><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, <a class="ref" href="yellow_3.html#T3">YELLOW_3:3</a></i>;<br/><a NAME="E9:41_4_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, </i>;<br/><a NAME="E10:41_4_1_7"/><b>then </b><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E12:41_4_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_8"><b>suppose </b></a><a NAME="E1:41_4_1_8"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_4_1_8"/><b>then </b><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E3:41_4_1_8"/><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_4_1_8"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:41_4_1_8"/><b>then </b><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><i><font color="Green">E123</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><a NAME="E7:41_4_1_8"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E1:41"><i><font color="Green">E29</font></i></a>, </i>;<br/><a NAME="E8:41_4_1_8"/><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/><a NAME="E9:41_4_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 ;<br/><a NAME="E10:41_4_1_8"/><b>then </b><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="yellow_3.html#T3">YELLOW_3:3</a></i>;<br/><a NAME="E11:41_4_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/><a NAME="E12:41_4_1_8"/><b>then </b><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, <a class="txt" href="lattice5.html#E6:41"><i><font color="Green">E42</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E14:41_4_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_9"><b>suppose </b></a><a NAME="E1:41_4_1_9"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:41_4_1_9"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E10:41"/><i><font color="Green">E131</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:41_5"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q</font>, <font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_5_1"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 </span><b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_5_1_1"><b>suppose </b></a><a NAME="E1:41_5_1_1"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><div class="add"><a NAME="E2:41_5_1_1"/><b>then </b><i><font color="Green">E134</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><a NAME="E3:41_5_1_1"/><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font>
 <b>by </b><i/>;<br/><a NAME="E4:41_5_1_1"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<font color="Maroon">q'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:41_5_1_1"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E6:41_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><a NAME="E7:41_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/><b>hence </b><a NAME="E8:41_5_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_5_1_2"><b>suppose </b></a><a NAME="E1:41_5_1_2"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><div class="add"><a NAME="E2:41_5_1_2"/><b>then </b><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E3:41_5_1_2"/><i><font color="Green">E138</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font>
 <b>by </b><i/>;<br/><a NAME="E4:41_5_1_2"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<font color="Maroon">q'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:41_5_1_2"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E6:41_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><a NAME="E7:41_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/><a NAME="E8:41_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/><a NAME="E9:41_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/><b>hence </b><a NAME="E10:41_5_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_5_1_3"><b>suppose </b></a><a NAME="E1:41_5_1_3"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><div class="add"><a NAME="E2:41_5_1_3"/><b>then </b><i><font color="Green">E140</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E3:41_5_1_3"/><i><font color="Green">E223</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font>
 <b>by </b><i/>;<br/><a NAME="E4:41_5_1_3"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<font color="Maroon">q'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:41_5_1_3"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E6:41_5_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><a NAME="E7:41_5_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/><b>hence </b><a NAME="E8:41_5_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E11:41"/><i><font color="Green">E224</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:41_6"/><i><font color="Green">E225</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_6_1"/>
( ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) )
 </span><b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_6_1_1"><b>suppose </b></a><a NAME="E1:41_6_1_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_6_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</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/><b>hence </b><a NAME="E3:41_6_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_6_1_2"><b>suppose </b></a><a NAME="E1:41_6_1_2"/><i><font color="Green">E226</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_6_1_2"/><b>then </b><i><font color="Green">E227</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E4:41_6_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_6_1_3"><b>suppose </b></a><a NAME="E1:41_6_1_3"/><i><font color="Green">E228</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_6_1_3"/><b>then </b><i><font color="Green">E229</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E4:41_6_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_6_1_4"><b>suppose </b></a><a NAME="E1:41_6_1_4"/><i><font color="Green">E230</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_6_1_4"/><b>then </b><i><font color="Green">E231</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E4:41_6_1_4"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_6_1_5"><b>suppose </b></a><a NAME="E1:41_6_1_5"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_6_1_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</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/><b>hence </b><a NAME="E3:41_6_1_5"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_6_1_6"><b>suppose </b></a><a NAME="E1:41_6_1_6"/><i><font color="Green">E232</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_6_1_6"/><b>then </b><i><font color="Green">E233</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E4:41_6_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_6_1_7"><b>suppose </b></a><a NAME="E1:41_6_1_7"/><i><font color="Green">E234</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_6_1_7"/><b>then </b><i><font color="Green">E235</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E4:41_6_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_6_1_8"><b>suppose </b></a><a NAME="E1:41_6_1_8"/><i><font color="Green">E236</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_6_1_8"/><b>then </b><i><font color="Green">E237</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E4:41_6_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_6_1_9"><b>suppose </b></a><a NAME="E1:41_6_1_9"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_6_1_9"/><b>then </b>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</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/><b>hence </b><a NAME="E3:41_6_1_9"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E12:41"/><i><font color="Green">E238</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:41_7"/><i><font color="Green">E239</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_7_1"/>
( ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_7_1_1"><b>suppose </b></a><a NAME="E1:41_7_1_1"/><i><font color="Green">E240</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font>
<b>by </b><i>, <a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:41_7_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_7_1_2"><b>suppose </b></a><a NAME="E1:41_7_1_2"/><i><font color="Green">E241</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E3:41_7_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_7_1_3"><b>suppose </b></a><a NAME="E1:41_7_1_3"/><i><font color="Green">E242</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_7_1_3"/><b>then </b><i><font color="Green">E243</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E3:41_7_1_3"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_7_1_3"/><b>then </b><i><font color="Green">E244</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><a NAME="E5:41_7_1_3"/><i><font color="Green">E245</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/><a NAME="E6:41_7_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E7:41_7_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:41"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow_3.html#T3">YELLOW_3:3</a></i>;<br/><a NAME="E8:41_7_1_3"/><b>then </b><i><font color="Green">E246</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, , , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E11:41_7_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E6:41"><i><font color="Green">E42</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_7_1_4"><b>suppose </b></a><a NAME="E1:41_7_1_4"/><i><font color="Green">E247</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
;<br/><b>hence </b><a NAME="E3:41_7_1_4"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_7_1_5"><b>suppose </b></a><a NAME="E1:41_7_1_5"/><i><font color="Green">E248</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font>
<b>by </b><i>, <a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:41_7_1_5"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_7_1_6"><b>suppose </b></a><a NAME="E1:41_7_1_6"/><i><font color="Green">E249</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_7_1_6"/><b>then </b><i><font color="Green">E250</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E3:41_7_1_6"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_7_1_6"/><b>then </b><i><font color="Green">E251</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><a NAME="E5:41_7_1_6"/><i><font color="Green">E252</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/><i><font color="Green">E253</font></i>: <font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><a NAME="E7:41_7_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E8:41_7_1_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:41"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow_3.html#T3">YELLOW_3:3</a></i>;<br/><a NAME="E9:41_7_1_6"/><b>then </b><i><font color="Green">E254</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, , , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E6:41"><i><font color="Green">E42</font></i></a>, <a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E11:41_7_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_7_1_7"><b>suppose </b></a><a NAME="E1:41_7_1_7"/><i><font color="Green">E255</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_7_1_7"/><b>then </b><i><font color="Green">E256</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><a NAME="E3:41_7_1_7"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_7_1_7"/><b>then </b><i><font color="Green">E257</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><i><font color="Green">E258</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/><i><font color="Green">E259</font></i>: <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><a NAME="E7:41_7_1_7"/>
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E8:41_7_1_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:41"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow_3.html#T3">YELLOW_3:3</a></i>;<br/><a NAME="E9:41_7_1_7"/><b>then </b><i><font color="Green">E260</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, , , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E6:41"><i><font color="Green">E42</font></i></a>, <a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E11:41_7_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_7_1_8"><b>suppose </b></a><a NAME="E1:41_7_1_8"/><i><font color="Green">E261</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E262</font></i>: <span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><a NAME="E3:41_7_1_8"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_7_1_8"/><b>then </b><i><font color="Green">E263</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><i><font color="Green">E264</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/><a NAME="E6:41_7_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E7:41_7_1_8"/><b>then </b><i><font color="Green">E265</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:41"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow_3.html#T3">YELLOW_3:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u'</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E9:41_7_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, , , <a class="txt" href="lattice5.html#E6:41"><i><font color="Green">E42</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_7_1_9"><b>suppose </b></a><a NAME="E1:41_7_1_9"/><i><font color="Green">E266</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font>
<b>by </b><i>, <a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:41_7_1_9"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E13:41"/><i><font color="Green">E267</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:41_8"/><i><font color="Green">E268</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_8_1"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> ) )
 </span><b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_1"><b>suppose </b></a><a NAME="E1:41_8_1_1"/><i><font color="Green">E269</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_8_1_1"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:41_8_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_2"><b>suppose </b></a><a NAME="E1:41_8_1_2"/><i><font color="Green">E270</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:41_8_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_3"><b>suppose </b></a><a NAME="E1:41_8_1_3"/><i><font color="Green">E271</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:41_8_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_4"><b>suppose </b></a><a NAME="E1:41_8_1_4"/><i><font color="Green">E272</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_8_1_4"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:41_8_1_4"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_5"><b>suppose </b></a><a NAME="E1:41_8_1_5"/><i><font color="Green">E273</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:41_8_1_5"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_6"><b>suppose </b></a><a NAME="E1:41_8_1_6"/><i><font color="Green">E274</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E3:41_8_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_7"><b>suppose </b></a><a NAME="E1:41_8_1_7"/><i><font color="Green">E275</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_8_1_7"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:41_8_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_8"><b>suppose </b></a><a NAME="E1:41_8_1_8"/><i><font color="Green">E276</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_8_1_8"/><b>then </b><i><font color="Green">E277</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
;<br/><b>hence </b><a NAME="E4:41_8_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_9"><b>suppose </b></a><a NAME="E1:41_8_1_9"/><i><font color="Green">E278</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E3:41_8_1_9"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_10"><b>suppose </b></a><a NAME="E1:41_8_1_10"/><i><font color="Green">E279</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E3:41_8_1_10"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_11"><b>suppose </b></a><a NAME="E1:41_8_1_11"/><i><font color="Green">E280</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_8_1_11"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:41_8_1_11"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_12"><b>suppose </b></a><a NAME="E1:41_8_1_12"/><i><font color="Green">E281</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_8_1_12"/><b>then </b><i><font color="Green">E282</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
;<br/><b>hence </b><a NAME="E4:41_8_1_12"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_13"><b>suppose </b></a><a NAME="E1:41_8_1_13"/><i><font color="Green">E283</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:41_8_1_13"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_14"><b>suppose </b></a><a NAME="E1:41_8_1_14"/><i><font color="Green">E284</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_8_1_14"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:41_8_1_14"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_15"><b>suppose </b></a><a NAME="E1:41_8_1_15"/><i><font color="Green">E285</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:41_8_1_15"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_16"><b>suppose </b></a><a NAME="E1:41_8_1_16"/><i><font color="Green">E286</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_8_1_16"/><b>then </b><i><font color="Green">E287</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
;<br/><b>hence </b><a NAME="E4:41_8_1_16"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_17"><b>suppose </b></a><a NAME="E1:41_8_1_17"/><i><font color="Green">E288</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_8_1_17"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:41_8_1_17"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_18"><b>suppose </b></a><a NAME="E1:41_8_1_18"/><i><font color="Green">E289</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E3:41_8_1_18"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_19"><b>suppose </b></a><a NAME="E1:41_8_1_19"/><i><font color="Green">E290</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E3:41_8_1_19"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_20"><b>suppose </b></a><a NAME="E1:41_8_1_20"/><i><font color="Green">E291</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_8_1_20"/><b>then </b><i><font color="Green">E292</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
;<br/><b>hence </b><a NAME="E4:41_8_1_20"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_21"><b>suppose </b></a><a NAME="E1:41_8_1_21"/><i><font color="Green">E293</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_8_1_21"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:41_8_1_21"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_22"><b>suppose </b></a><a NAME="E1:41_8_1_22"/><i><font color="Green">E294</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E3:41_8_1_22"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_23"><b>suppose </b></a><a NAME="E1:41_8_1_23"/><i><font color="Green">E295</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E3:41_8_1_23"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_24"><b>suppose </b></a><a NAME="E1:41_8_1_24"/><i><font color="Green">E296</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_8_1_24"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:41_8_1_24"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_25"><b>suppose </b></a><a NAME="E1:41_8_1_25"/><i><font color="Green">E297</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:41_8_1_25"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_26"><b>suppose </b></a><a NAME="E1:41_8_1_26"/><i><font color="Green">E298</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:41_8_1_26"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_8_1_27"><b>suppose </b></a><a NAME="E1:41_8_1_27"/><i><font color="Green">E299</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:41_8_1_27"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:41_8_1_27"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E14:41"/>
for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font> holds  <span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice5.html#K6" target="_self">new_set</a> <font color="Maroon">A</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_9_1"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) )
 </span><b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_9_1_1"><b>suppose </b></a><a NAME="E1:41_9_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_9_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_9_1_2"><b>suppose </b></a><a NAME="E1:41_9_1_2"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_9_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_9_1_3"><b>suppose </b></a><a NAME="E1:41_9_1_3"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_9_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_9_1_4"><b>suppose </b></a><a NAME="E1:41_9_1_4"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_9_1_4"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_9_1_5"><b>suppose </b></a><a NAME="E1:41_9_1_5"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_9_1_5"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_9_1_6"><b>suppose </b></a><a NAME="E1:41_9_1_6"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_9_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_9_1_7"><b>suppose </b></a><a NAME="E1:41_9_1_7"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_9_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_9_1_8"><b>suppose </b></a><a NAME="E1:41_9_1_8"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_9_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E2:41"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E15:41"/>
 <a href="lattice5.html#K7" target="_self">new_bi_fun</a> <font color="Maroon">d</font>,<font color="Maroon">q</font> is <a href="lattice5.html#V3" target="_self">u.t.i.</a>
 <b>by </b><i/>;<br/>


</div>
