<?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>assume </b><a NAME="E1:22"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">L</font> is <a href="yellow11.html#V1">modular</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="E2:22"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">d</font> is <a href="lattice5.html#V1">symmetric</a>
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">d</font> is <a href="lattice5.html#V3">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>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">a</font> = <font color="Maroon">q</font> <a href="mcart_1.html#K10">`3</a> ;<br/>
<b>set </b><font color="Maroon">b</font> = <font color="Maroon">q</font> <a href="mcart_1.html#K11">`4</a> ;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font>;<br/>
<b>assume </b><a NAME="E4:22"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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>reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="tarski.html#K2">{</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><a href="tarski.html#K2">}</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="E7:22"/><i><font color="Green">E60</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="lattice8.html#K1" target="_self">new_set2</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_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="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:22_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="E3:22_1"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</a> <font color="Maroon">p'</font>,<font color="Maroon">u'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:22_1"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:22_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:22_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:22"/><i><font color="Green">E63</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="lattice8.html#K1" target="_self">new_set2</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_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="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:22_2"/><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">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="22_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_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> ) )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1"><b>suppose </b></a><a NAME="E1:22_2_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">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="E3:22_2_1_1"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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="E4:22_2_1_1"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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="E5:22_2_1_1"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 <b>by </b><i/>;<br/><a NAME="E6:22_2_1_1"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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>, <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/><a NAME="E7:22_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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="E8:22_2_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_2_1_2"><b>suppose </b></a><a NAME="E1:22_2_1_2"/><i><font color="Green">E75</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="E3:22_2_1_2"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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>
 <b>by </b><i>, </i>;<br/><a NAME="E4:22_2_1_2"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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">a</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:22_2_1_2"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 <b>by </b><i/>;<br/><a NAME="E6:22_2_1_2"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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>, <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/><a NAME="E7:22_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">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="E8:22_2_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E9:22"/><i><font color="Green">E80</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="lattice8.html#K1" target="_self">new_set2</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_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="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:22_3"/><i><font color="Green">E81</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="22_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_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> ) )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_3_1_1"><b>suppose </b></a><a NAME="E1:22_3_1_1"/><i><font color="Green">E82</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="E3:22_3_1_1"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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="E4:22_3_1_1"/><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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="E5:22_3_1_1"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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">.</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">E86</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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">.</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">.</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">.</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">.</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">.</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">.</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">.</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="E7:22_3_1_1"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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>, <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/><a NAME="E8:22_3_1_1"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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>, <a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a></i>;<br/><a NAME="E9:22_3_1_1"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E10:22_3_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_3_1_2"><b>suppose </b></a><a NAME="E1:22_3_1_2"/><i><font color="Green">E87</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="E3:22_3_1_2"/><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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>
 <b>by </b><i>, </i>;<br/><a NAME="E4:22_3_1_2"/><i><font color="Green">E89</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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">a</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:22_3_1_2"/><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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">.</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">a</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><i><font color="Green">E91</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">d</font> <a href="lattice5.html#K4">.</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">.</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">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">.</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">.</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">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">.</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">d</font> <a href="lattice5.html#K4">.</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/><a NAME="E7:22_3_1_2"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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>, <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/><a NAME="E8:22_3_1_2"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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="lattice5.html#D6">LATTICE5:def 6</a></i>;<br/><a NAME="E9:22_3_1_2"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">d</font> <a href="lattice5.html#K4">.</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="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E10:22_3_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E10:22"/><i><font color="Green">E92</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="lattice8.html#K1" target="_self">new_set2</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_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="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:22_4"/><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#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="22_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_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"><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> ) )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1"><b>suppose </b></a><a NAME="E1:22_4_1_1"/><i><font color="Green">E94</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E3:22_4_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_4_1_2"><b>suppose </b></a><a NAME="E1:22_4_1_2"/><i><font color="Green">E95</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 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/><a NAME="E3:22_4_1_2"/><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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="E4:22_4_1_2"/><i><font color="Green">E126</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:22_4_1_2"/>
<font color="Maroon">a</font> <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">.</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="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E6:22_4_1_2"/><b>then </b><i><font color="Green">E127</font></i>: 
<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">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</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#K12">"/\"</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">.</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>
 <b>by </b><i>, <a class="ref" href="yellow11.html#D3">YELLOW11:def 3</a></i>;<br/><a NAME="E7:22_4_1_2"/><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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="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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_5.html#T2">YELLOW_5:2</a></i>;<br/><a NAME="E8:22_4_1_2"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E9:22_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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="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">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="yellow_3.html#T3">YELLOW_3:3</a></i>;<br/><a NAME="E10:22_4_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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</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="ref" href="yellow_5.html#T6">YELLOW_5:6</a></i>;<br/><a NAME="E11:22_4_1_2"/><b>then </b><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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></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">.</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">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</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> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">b</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <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">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></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">.</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">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</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">a</font></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="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">.</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">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</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> <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/><a NAME="E13:22_4_1_2"/><b>then </b><i><font color="Green">E130</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">a</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E14:22_4_1_2"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/><a NAME="E15:22_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">a</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T7">YELLOW_5:7</a></i>;<br/><a NAME="E16:22_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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"><span class="p2">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E17:22_4_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_4_1_3"><b>suppose </b></a><a NAME="E1:22_4_1_3"/><i><font color="Green">E131</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 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/><a NAME="E3:22_4_1_3"/><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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>
 <b>by </b><i>, </i>;<br/><a NAME="E4:22_4_1_3"/><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:22_4_1_3"/>
<font color="Maroon">a</font> <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">.</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="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E6:22_4_1_3"/><b>then </b><i><font color="Green">E134</font></i>: 
<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">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</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#K12">"/\"</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">.</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>
 <b>by </b><i>, <a class="ref" href="yellow11.html#D3">YELLOW11:def 3</a></i>;<br/><a NAME="E7:22_4_1_3"/><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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="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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_5.html#T2">YELLOW_5:2</a></i>;<br/><a NAME="E8:22_4_1_3"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E9:22_4_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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="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">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="yellow_3.html#T3">YELLOW_3:3</a></i>;<br/><a NAME="E10:22_4_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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</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="ref" href="yellow_5.html#T6">YELLOW_5:6</a></i>;<br/><a NAME="E11:22_4_1_3"/><b>then </b><i><font color="Green">E136</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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></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">.</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">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</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> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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#K12">"/\"</a> <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">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></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">.</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">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</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">a</font></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="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">.</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">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</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> <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/><a NAME="E13:22_4_1_3"/><b>then </b><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">a</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E14:22_4_1_3"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="lattice5.html#K4">.</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>
 <b>by </b><i>, <a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a></i>;<br/><a NAME="E15:22_4_1_3"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/><a NAME="E16:22_4_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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">a</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T7">YELLOW_5:7</a></i>;<br/><a NAME="E17:22_4_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E18:22_4_1_3"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_4_1_4"><b>suppose </b></a><a NAME="E1:22_4_1_4"/><i><font color="Green">E138</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E3:22_4_1_4"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E11:22"/><i><font color="Green">E139</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="lattice8.html#K1" target="_self">new_set2</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_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="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:22_5"/><i><font color="Green">E140</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="22_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_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> ) )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_5_1_1"><b>suppose </b></a><a NAME="E1:22_5_1_1"/><i><font color="Green">E141</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="E2:22_5_1_1"/><b>then </b><i><font color="Green">E161</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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="E3:22_5_1_1"/><i><font color="Green">E162</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font>
 <b>by </b><i/>;<br/><a NAME="E4:22_5_1_1"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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>, <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/><a NAME="E5:22_5_1_1"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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>, <a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a></i>;<br/><a NAME="E6:22_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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="E7:22_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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="E8:22_5_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_5_1_2"><b>suppose </b></a><a NAME="E1:22_5_1_2"/><i><font color="Green">E163</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="E2:22_5_1_2"/><b>then </b><i><font color="Green">E164</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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">a</font>
 <b>by </b><i/>;<br/><a NAME="E3:22_5_1_2"/><i><font color="Green">E165</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font>
 <b>by </b><i/>;<br/><a NAME="E4:22_5_1_2"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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>, <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/><a NAME="E5:22_5_1_2"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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>, <a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a></i>;<br/><a NAME="E6:22_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">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">.</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">.</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="E7:22_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">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">.</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">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">.</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="E8:22_5_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E12:22"/><i><font color="Green">E166</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="lattice8.html#K1" target="_self">new_set2</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_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="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:22_6"/><i><font color="Green">E167</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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_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"><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> ) )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_6_1_1"><b>suppose </b></a><a NAME="E1:22_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="E2:22_6_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E3:22_6_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_6_1_2"><b>suppose </b></a><a NAME="E1:22_6_1_2"/><i><font color="Green">E168</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"><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/><a NAME="E3:22_6_1_2"/><i><font color="Green">E169</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:22_6_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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/><b>then </b><i><font color="Green">E170</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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">.</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">a</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">a</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">.</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> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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></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>
<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">.</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">.</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">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">.</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">.</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">a</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
;<br/><a NAME="E6:22_6_1_2"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">q'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<font color="Maroon">q'</font>
 <b>by </b><i>, <a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a></i>;<br/><a NAME="E7:22_6_1_2"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">d</font> <a href="lattice5.html#K4">.</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">d</font> <a href="lattice5.html#K4">.</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>, <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/><a NAME="E8:22_6_1_2"/><b>then </b><i><font color="Green">E171</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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="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"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_5.html#T7">YELLOW_5:7</a></i>;<br/><a NAME="E9:22_6_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</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">.</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="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><b>hence </b><a NAME="E10:22_6_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, , <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="22_6_1_3"><b>suppose </b></a><a NAME="E1:22_6_1_3"/><i><font color="Green">E172</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"><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/><a NAME="E3:22_6_1_3"/><i><font color="Green">E173</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:22_6_1_3"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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">a</font>
 <b>by </b><i>, </i>;<br/><b>then </b><i><font color="Green">E174</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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">.</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">a</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">a</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">.</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> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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></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>
<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">.</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">.</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">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">.</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">.</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">a</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
;<br/><a NAME="E6:22_6_1_3"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">q'</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<font color="Maroon">q'</font>
 <b>by </b><i>, <a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a></i>;<br/><a NAME="E7:22_6_1_3"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">d</font> <a href="lattice5.html#K4">.</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">d</font> <a href="lattice5.html#K4">.</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>, <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/><a NAME="E8:22_6_1_3"/><b>then </b><i><font color="Green">E175</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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="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"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_5.html#T7">YELLOW_5:7</a></i>;<br/><a NAME="E9:22_6_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</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">.</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="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><b>hence </b><a NAME="E10:22_6_1_3"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, , <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="22_6_1_4"><b>suppose </b></a><a NAME="E1:22_6_1_4"/>
( <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="E2:22_6_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E3:22_6_1_4"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E13:22"/><i><font color="Green">E176</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="lattice8.html#K1" target="_self">new_set2</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_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="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:22_7"/><i><font color="Green">E177</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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_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"><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> ) )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_7_1_1"><b>suppose </b></a><a NAME="E1:22_7_1_1"/><i><font color="Green">E178</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E3:22_7_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_7_1_2"><b>suppose </b></a><a NAME="E1:22_7_1_2"/><i><font color="Green">E179</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 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/><a NAME="E3:22_7_1_2"/><i><font color="Green">E180</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:22_7_1_2"/><i><font color="Green">E181</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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="E5:22_7_1_2"/>
<font color="Maroon">a</font> <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">.</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="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E6:22_7_1_2"/><b>then </b><i><font color="Green">E182</font></i>: 
<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">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</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#K12">"/\"</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">.</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>
 <b>by </b><i>, <a class="ref" href="yellow11.html#D3">YELLOW11:def 3</a></i>;<br/><a NAME="E7:22_7_1_2"/><i><font color="Green">E183</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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="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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_5.html#T2">YELLOW_5:2</a></i>;<br/><a NAME="E8:22_7_1_2"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E9:22_7_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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="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">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="yellow_3.html#T3">YELLOW_3:3</a></i>;<br/><a NAME="E10:22_7_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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</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="ref" href="yellow_5.html#T6">YELLOW_5:6</a></i>;<br/><a NAME="E11:22_7_1_2"/><b>then </b><i><font color="Green">E184</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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></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">.</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"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</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> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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#K12">"/\"</a> <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">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></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">.</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"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</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">a</font></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="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">.</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"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</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> <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/><a NAME="E13:22_7_1_2"/><b>then </b><i><font color="Green">E185</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">a</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E14:22_7_1_2"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/><a NAME="E15:22_7_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">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">.</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">.</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">a</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T7">YELLOW_5:7</a></i>;<br/><a NAME="E16:22_7_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">a</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E17:22_7_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_7_1_3"><b>suppose </b></a><a NAME="E1:22_7_1_3"/><i><font color="Green">E186</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 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/><a NAME="E3:22_7_1_3"/><i><font color="Green">E187</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:22_7_1_3"/><i><font color="Green">E188</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</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">a</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:22_7_1_3"/>
<font color="Maroon">a</font> <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">.</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="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E6:22_7_1_3"/><b>then </b><i><font color="Green">E189</font></i>: 
<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">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</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#K12">"/\"</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">.</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>
 <b>by </b><i>, <a class="ref" href="yellow11.html#D3">YELLOW11:def 3</a></i>;<br/><a NAME="E7:22_7_1_3"/><i><font color="Green">E190</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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="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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_5.html#T2">YELLOW_5:2</a></i>;<br/><a NAME="E8:22_7_1_3"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E9:22_7_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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="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">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="yellow_3.html#T3">YELLOW_3:3</a></i>;<br/><a NAME="E10:22_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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</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">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</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="ref" href="yellow_5.html#T6">YELLOW_5:6</a></i>;<br/><a NAME="E11:22_7_1_3"/><b>then </b><i><font color="Green">E191</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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></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">.</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</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> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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"><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#K12">"/\"</a> <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">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></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">.</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</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">a</font></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="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">.</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</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> <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/><a NAME="E13:22_7_1_3"/><b>then </b><i><font color="Green">E192</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">a</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E14:22_7_1_3"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="lattice5.html#K4">.</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>
 <b>by </b><i>, <a class="ref" href="lattice5.html#D6">LATTICE5:def 6</a></i>;<br/><a NAME="E15:22_7_1_3"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/><a NAME="E16:22_7_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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">.</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">.</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">a</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T7">YELLOW_5:7</a></i>;<br/><a NAME="E17:22_7_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">q</font>,<font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E18:22_7_1_3"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_7_1_4"><b>suppose </b></a><a NAME="E1:22_7_1_4"/><i><font color="Green">E193</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E3:22_7_1_4"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E14:22"/><i><font color="Green">E194</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="lattice8.html#K1" target="_self">new_set2</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_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="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:22_8"/><i><font color="Green">E195</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="22_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_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"><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"><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"><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> ) )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_8_1_1"><b>suppose </b></a><a NAME="E1:22_8_1_1"/><i><font color="Green">E196</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="E2:22_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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E3:22_8_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_8_1_2"><b>suppose </b></a><a NAME="E1:22_8_1_2"/><i><font color="Green">E197</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</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="E3:22_8_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_8_1_3"><b>suppose </b></a><a NAME="E1:22_8_1_3"/><i><font color="Green">E198</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="E2:22_8_1_3"/>
 <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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E3:22_8_1_3"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_8_1_4"><b>suppose </b></a><a NAME="E1:22_8_1_4"/><i><font color="Green">E199</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</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="E3:22_8_1_4"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_8_1_5"><b>suppose </b></a><a NAME="E1:22_8_1_5"/><i><font color="Green">E200</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E3:22_8_1_5"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_8_1_6"><b>suppose </b></a><a NAME="E1:22_8_1_6"/><i><font color="Green">E201</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="E2:22_8_1_6"/>
 <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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E3:22_8_1_6"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_8_1_7"><b>suppose </b></a><a NAME="E1:22_8_1_7"/><i><font color="Green">E202</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<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#K12">"/\"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</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="E3:22_8_1_7"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_8_1_8"><b>suppose </b></a><a NAME="E1:22_8_1_8"/><i><font color="Green">E203</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="E2:22_8_1_8"/>
 <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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E3:22_8_1_8"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="E15:22"/>
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="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font> holds  <span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Olive">q</font>,<font color="Olive">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_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="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>, <font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K1" target="_self">new_set2</a> <font color="Maroon">A</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_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="22_9_1_1"><b>suppose </b></a><a NAME="E1:22_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="E2:22_9_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_9_1_2"><b>suppose </b></a><a NAME="E1:22_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="E2:22_9_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_9_1_3"><b>suppose </b></a><a NAME="E1:22_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="E2:22_9_1_3"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_9_1_4"><b>suppose </b></a><a NAME="E1:22_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="E2:22_9_1_4"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_9_1_5"><b>suppose </b></a><a NAME="E1:22_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="E2:22_9_1_5"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_9_1_6"><b>suppose </b></a><a NAME="E1:22_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="E2:22_9_1_6"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_9_1_7"><b>suppose </b></a><a NAME="E1:22_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="E2:22_9_1_7"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="22_9_1_8"><b>suppose </b></a><a NAME="E1:22_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="E2:22_9_1_8"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font></span>)</span> <a href="lattice5.html#K4">.</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>
<b>hence </b><a NAME="E16:22"/>
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <font color="Maroon">d</font>,<font color="Maroon">q</font> is <a href="lattice5.html#V3">u.t.i.</a>
 <b>by </b><i><a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/>


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