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

<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/>



<b>assume </b><a NAME="E1:22"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c2">L</font> is <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">d</font> be   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:22"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c3">d</font> is <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c3">d</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">q</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>;<br/>

<b>set </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> ;<br/>
<b>set </b><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> ;<br/>
<b>set </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> ;<br/>
<b>set </b><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> ;<br/>
<b>set </b><font color="Maroon" title="c9">f</font> =  <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font>;<br/>
<b>assume </b><a NAME="E4:22"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> </span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">B</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">a</font> = <font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> , <font color="Maroon" title="c12">b</font> = <font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> ;<br/>
<a NAME="E7:22"/><i><font color="Green" title="E20">A5</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">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" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E1:22_1"/>
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c16">p'</font> = <font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c17">q'</font> = <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c18">u'</font> = <font color="Maroon" title="c15">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<a NAME="E3:22_1"/><i><font color="Green" title="E21">A6</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c18">u'</font>
 
<b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/>
<a NAME="E4:22_1"/><i><font color="Green" title="E22">A7</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c17">q'</font>
 
<b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/>
<a NAME="E5:22_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">q'</font>,<font color="Maroon" title="c18">u'</font>
 
<b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E6:22_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="lattice8.html#E3:22_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="lattice8.html#E4:22_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:22"/><i><font color="Green" title="E21">A8</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">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" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E1:22_2"/><i><font color="Green" title="E22">A9</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">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" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E1:22_2"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">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" title="E23">A10</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c16">p'</font> = <font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c17">q'</font> = <font color="Maroon" title="c14">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:22_2_1_1"/><i><font color="Green" title="E24">A11</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_2_1_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E4:22_2_1_1"/><i><font color="Green" title="E25">A12</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_2_1_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E5:22_2_1_1"/><i><font color="Green" title="E26">A13</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c17">q'</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E6:22_2_1_1"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c17">q'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c17">q'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22_2_1_1"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="lattice8.html#E4:22_2_1_1"><i><font color="Green" title="E25">A12</font></i></a>, <a class="txt" href="lattice8.html#E5:22_2_1_1"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">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" title="E23">A14</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c16">p'</font> = <font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c17">q'</font> = <font color="Maroon" title="c14">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:22_2_1_2"/><i><font color="Green" title="E24">A15</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_2_1_2"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E4:22_2_1_2"/><i><font color="Green" title="E25">A16</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_2_1_2"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E5:22_2_1_2"/><i><font color="Green" title="E26">A17</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c17">q'</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E6:22_2_1_2"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c17">q'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c17">q'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22_2_1_2"><i><font color="Green" title="E24">A15</font></i></a>, <a class="txt" href="lattice8.html#E4:22_2_1_2"><i><font color="Green" title="E25">A16</font></i></a>, <a class="txt" href="lattice8.html#E5:22_2_1_2"><i><font color="Green" title="E26">A17</font></i></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">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" title="E22">A18</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">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" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E1:22_3"/><i><font color="Green" title="E23">A19</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E1:22_3"><i><font color="Green" title="E23">A19</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">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" title="E24">A20</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c16">p'</font> = <font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c17">u'</font> = <font color="Maroon" title="c15">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:22_3_1_1"/><i><font color="Green" title="E25">A21</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_3_1_1"><i><font color="Green" title="E24">A20</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E4:22_3_1_1"/><i><font color="Green" title="E26">A22</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_3_1_1"><i><font color="Green" title="E24">A20</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E5:22_3_1_1"/><i><font color="Green" title="E27">A23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><i><font color="Green" title="E28">A24</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><a NAME="E7:22_3_1_1"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c17">u'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<font color="Maroon" title="c17">u'</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a></i>;<br/><a NAME="E8:22_3_1_1"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c17">u'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:22"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="lattice5.html#D6" title="LATTICE5:def.6">LATTICE5:def 6</a></i>;<br/><a NAME="E9:22_3_1_1"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c17">u'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E5:22_3_1_1"><i><font color="Green" title="E27">A23</font></i></a>, <a class="txt" href="lattice8.html#E6:22_3_1_1"><i><font color="Green" title="E28">A24</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22_3_1_1"><i><font color="Green" title="E25">A21</font></i></a>, <a class="txt" href="lattice8.html#E4:22_3_1_1"><i><font color="Green" title="E26">A22</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></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" title="E24">A25</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c16">p'</font> = <font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c17">u'</font> = <font color="Maroon" title="c15">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:22_3_1_2"/><i><font color="Green" title="E25">A26</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_3_1_2"><i><font color="Green" title="E24">A25</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E4:22_3_1_2"/><i><font color="Green" title="E26">A27</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_3_1_2"><i><font color="Green" title="E24">A25</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E5:22_3_1_2"/><i><font color="Green" title="E27">A28</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><i><font color="Green" title="E28">A29</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><a NAME="E7:22_3_1_2"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c17">u'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<font color="Maroon" title="c17">u'</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a></i>;<br/><a NAME="E8:22_3_1_2"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c17">u'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:22"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="lattice5.html#D6" title="LATTICE5:def.6">LATTICE5:def 6</a></i>;<br/><a NAME="E9:22_3_1_2"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<font color="Maroon" title="c17">u'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E5:22_3_1_2"><i><font color="Green" title="E27">A28</font></i></a>, <a class="txt" href="lattice8.html#E6:22_3_1_2"><i><font color="Green" title="E28">A29</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22_3_1_2"><i><font color="Green" title="E25">A26</font></i></a>, <a class="txt" href="lattice8.html#E4:22_3_1_2"><i><font color="Green" title="E26">A27</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></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" title="E23">A30</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">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" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E1:22_4"/><i><font color="Green" title="E24">A31</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">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" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E1:22_4"><i><font color="Green" title="E24">A31</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">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" title="E25">A32</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_4_1_1"><i><font color="Green" title="E25">A32</font></i></a></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" title="E25">A33</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c16">p'</font> = <font color="Maroon" title="c13">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:22_4_1_2"/><i><font color="Green" title="E26">A34</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_4_1_2"><i><font color="Green" title="E25">A33</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E4:22_4_1_2"/><i><font color="Green" title="E27">A35</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_4_1_2"><i><font color="Green" title="E25">A33</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E5:22_4_1_2"/>
<font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><a NAME="E6:22_4_1_2"/><b>then </b><i><font color="Green" title="E28">A36</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="yellow11.html#D3" title="YELLOW11:def.3">YELLOW11:def 3</a></i>;<br/><a NAME="E7:22_4_1_2"/><i><font color="Green" title="E29">A37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_5.html#T2" title="YELLOW_5:th.2">YELLOW_5:2</a></i>;<br/><a NAME="E8:22_4_1_2"/>
<font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">a</font>
 ;<br/><a NAME="E9:22_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:22"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_5.html#T6" title="YELLOW_5:th.6">YELLOW_5:6</a></i>;<br/><a NAME="E11:22_4_1_2"/><b>then </b><i><font color="Green" title="E30">A38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E3:22_4_1_2"><i><font color="Green" title="E26">A34</font></i></a>, <a class="txt" href="lattice8.html#E4:22_4_1_2"><i><font color="Green" title="E27">A35</font></i></a>, <a class="txt" href="lattice8.html#E6:22_4_1_2"><i><font color="Green" title="E28">A36</font></i></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><a NAME="E13:22_4_1_2"/><b>then </b><i><font color="Green" title="E31">A39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E7:22_4_1_2"><i><font color="Green" title="E29">A37</font></i></a>, <a class="txt" href="lattice8.html#E11:22_4_1_2"><i><font color="Green" title="E30">A38</font></i></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><a NAME="E14:22_4_1_2"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T7" title="YELLOW_5:th.7">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E13:22_4_1_2"><i><font color="Green" title="E31">A39</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_4_1_2"><i><font color="Green" title="E25">A33</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></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" title="E25">A40</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c16">p'</font> = <font color="Maroon" title="c13">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:22_4_1_3"/><i><font color="Green" title="E26">A41</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_4_1_3"><i><font color="Green" title="E25">A40</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E4:22_4_1_3"/><i><font color="Green" title="E27">A42</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_4_1_3"><i><font color="Green" title="E25">A40</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E5:22_4_1_3"/>
<font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><a NAME="E6:22_4_1_3"/><b>then </b><i><font color="Green" title="E28">A43</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="yellow11.html#D3" title="YELLOW11:def.3">YELLOW11:def 3</a></i>;<br/><a NAME="E7:22_4_1_3"/><i><font color="Green" title="E29">A44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_5.html#T2" title="YELLOW_5:th.2">YELLOW_5:2</a></i>;<br/><a NAME="E8:22_4_1_3"/>
<font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">a</font>
 ;<br/><a NAME="E9:22_4_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:22"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_5.html#T6" title="YELLOW_5:th.6">YELLOW_5:6</a></i>;<br/><a NAME="E11:22_4_1_3"/><b>then </b><i><font color="Green" title="E30">A45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E3:22_4_1_3"><i><font color="Green" title="E26">A41</font></i></a>, <a class="txt" href="lattice8.html#E4:22_4_1_3"><i><font color="Green" title="E27">A42</font></i></a>, <a class="txt" href="lattice8.html#E6:22_4_1_3"><i><font color="Green" title="E28">A43</font></i></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><a NAME="E13:22_4_1_3"/><b>then </b><i><font color="Green" title="E31">A46</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E7:22_4_1_3"><i><font color="Green" title="E29">A44</font></i></a>, <a class="txt" href="lattice8.html#E11:22_4_1_3"><i><font color="Green" title="E30">A45</font></i></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><a NAME="E14:22_4_1_3"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:22"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="lattice5.html#D6" title="LATTICE5:def.6">LATTICE5:def 6</a></i>;<br/><a NAME="E15:22_4_1_3"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T7" title="YELLOW_5:th.7">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E13:22_4_1_3"><i><font color="Green" title="E31">A46</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_4_1_3"><i><font color="Green" title="E25">A40</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></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" title="E25">A47</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_4_1_4"><i><font color="Green" title="E25">A47</font></i></a></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" title="E24">A48</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">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" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E1:22_5"/><i><font color="Green" title="E25">A49</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c16">q'</font> = <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c17">u'</font> = <font color="Maroon" title="c15">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">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" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E1:22_5"><i><font color="Green" title="E25">A49</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">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" title="E26">A50</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ;<br/><div class="add"><a NAME="E2:22_5_1_1"/><b>then </b><i><font color="Green" title="E27">A51</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E3:22_5_1_1"/><i><font color="Green" title="E28">A52</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<font color="Maroon" title="c17">u'</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E4:22_5_1_1"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<font color="Maroon" title="c16">q'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a></i>;<br/><a NAME="E5:22_5_1_1"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<font color="Maroon" title="c17">u'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:22"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="lattice5.html#D6" title="LATTICE5:def.6">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<font color="Maroon" title="c17">u'</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<font color="Maroon" title="c17">u'</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_5_1_1"><i><font color="Green" title="E26">A50</font></i></a>, <a class="txt" href="lattice8.html#E2:22_5_1_1"><i><font color="Green" title="E27">A51</font></i></a>, <a class="txt" href="lattice8.html#E3:22_5_1_1"><i><font color="Green" title="E28">A52</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></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" title="E26">A53</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ;<br/><div class="add"><a NAME="E2:22_5_1_2"/><b>then </b><i><font color="Green" title="E27">A54</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E3:22_5_1_2"/><i><font color="Green" title="E28">A55</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<font color="Maroon" title="c17">u'</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E4:22_5_1_2"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<font color="Maroon" title="c16">q'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a></i>;<br/><a NAME="E5:22_5_1_2"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<font color="Maroon" title="c17">u'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:22"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="lattice5.html#D6" title="LATTICE5:def.6">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<font color="Maroon" title="c17">u'</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c17">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<font color="Maroon" title="c17">u'</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_5_1_2"><i><font color="Green" title="E26">A53</font></i></a>, <a class="txt" href="lattice8.html#E2:22_5_1_2"><i><font color="Green" title="E27">A54</font></i></a>, <a class="txt" href="lattice8.html#E3:22_5_1_2"><i><font color="Green" title="E28">A55</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></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" title="E25">A56</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">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" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E1:22_6"/><i><font color="Green" title="E26">A57</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">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" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E1:22_6"><i><font color="Green" title="E26">A57</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">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" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E3:22_6_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">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" title="E27">A58</font></i>: 
( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c16">q'</font> = <font color="Maroon" title="c14">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:22_6_1_2"/><i><font color="Green" title="E28">A59</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_6_1_2"><i><font color="Green" title="E27">A58</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E4:22_6_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_6_1_2"><i><font color="Green" title="E27">A58</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><b>then </b><i><font color="Green" title="E29">A60</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E1:22_6_1_2"><i><font color="Green" title="E27">A58</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
;<br/><a NAME="E6:22_6_1_2"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<font color="Maroon" title="c16">q'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:22"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="lattice5.html#D6" title="LATTICE5:def.6">LATTICE5:def 6</a></i>;<br/><a NAME="E7:22_6_1_2"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a></i>;<br/><a NAME="E8:22_6_1_2"/><b>then </b><i><font color="Green" title="E30">A61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E5:22_6_1_2"><i><font color="Green" title="E29">A60</font></i></a>, <a class="ref" href="yellow_5.html#T7" title="YELLOW_5:th.7">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22_6_1_2"><i><font color="Green" title="E28">A59</font></i></a>, <a class="txt" href="lattice8.html#E8:22_6_1_2"><i><font color="Green" title="E30">A61</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">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" title="E27">A62</font></i>: 
( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c16">q'</font> = <font color="Maroon" title="c14">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:22_6_1_3"/><i><font color="Green" title="E28">A63</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_6_1_3"><i><font color="Green" title="E27">A62</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E4:22_6_1_3"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_6_1_3"><i><font color="Green" title="E27">A62</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><b>then </b><i><font color="Green" title="E29">A64</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E1:22_6_1_3"><i><font color="Green" title="E27">A62</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
;<br/><a NAME="E6:22_6_1_3"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<font color="Maroon" title="c16">q'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:22"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="lattice5.html#D6" title="LATTICE5:def.6">LATTICE5:def 6</a></i>;<br/><a NAME="E7:22_6_1_3"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a></i>;<br/><a NAME="E8:22_6_1_3"/><b>then </b><i><font color="Green" title="E30">A65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E5:22_6_1_3"><i><font color="Green" title="E29">A64</font></i></a>, <a class="ref" href="yellow_5.html#T7" title="YELLOW_5:th.7">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22_6_1_3"><i><font color="Green" title="E28">A63</font></i></a>, <a class="txt" href="lattice8.html#E8:22_6_1_3"><i><font color="Green" title="E30">A65</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">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" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E3:22_6_1_4"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">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" title="E26">A66</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">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" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E1:22_7"/><i><font color="Green" title="E27">A67</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E1:22_7"><i><font color="Green" title="E27">A67</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">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" title="E28">A68</font></i>: 
( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:22_7_1_1"><i><font color="Green" title="E28">A68</font></i></a>, <a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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" title="E28">A69</font></i>: 
( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c16">u'</font> = <font color="Maroon" title="c15">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:22_7_1_2"/><i><font color="Green" title="E29">A70</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_7_1_2"><i><font color="Green" title="E28">A69</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E4:22_7_1_2"/><i><font color="Green" title="E30">A71</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_7_1_2"><i><font color="Green" title="E28">A69</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E5:22_7_1_2"/>
<font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><a NAME="E6:22_7_1_2"/><b>then </b><i><font color="Green" title="E31">A72</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="yellow11.html#D3" title="YELLOW11:def.3">YELLOW11:def 3</a></i>;<br/><a NAME="E7:22_7_1_2"/><i><font color="Green" title="E32">A73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_5.html#T2" title="YELLOW_5:th.2">YELLOW_5:2</a></i>;<br/><a NAME="E8:22_7_1_2"/>
<font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">a</font>
 ;<br/><a NAME="E9:22_7_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:22"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_5.html#T6" title="YELLOW_5:th.6">YELLOW_5:6</a></i>;<br/><a NAME="E11:22_7_1_2"/><b>then </b><i><font color="Green" title="E33">A74</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E3:22_7_1_2"><i><font color="Green" title="E29">A70</font></i></a>, <a class="txt" href="lattice8.html#E4:22_7_1_2"><i><font color="Green" title="E30">A71</font></i></a>, <a class="txt" href="lattice8.html#E6:22_7_1_2"><i><font color="Green" title="E31">A72</font></i></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><a NAME="E13:22_7_1_2"/><b>then </b><i><font color="Green" title="E34">A75</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E7:22_7_1_2"><i><font color="Green" title="E32">A73</font></i></a>, <a class="txt" href="lattice8.html#E11:22_7_1_2"><i><font color="Green" title="E33">A74</font></i></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><a NAME="E14:22_7_1_2"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T7" title="YELLOW_5:th.7">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E13:22_7_1_2"><i><font color="Green" title="E34">A75</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_7_1_2"><i><font color="Green" title="E28">A69</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></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" title="E28">A76</font></i>: 
( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c16">u'</font> = <font color="Maroon" title="c15">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:22_7_1_3"/><i><font color="Green" title="E29">A77</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_7_1_3"><i><font color="Green" title="E28">A76</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E4:22_7_1_3"/><i><font color="Green" title="E30">A78</font></i>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_7_1_3"><i><font color="Green" title="E28">A76</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>;<br/><a NAME="E5:22_7_1_3"/>
<font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><a NAME="E6:22_7_1_3"/><b>then </b><i><font color="Green" title="E31">A79</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="yellow11.html#D3" title="YELLOW11:def.3">YELLOW11:def 3</a></i>;<br/><a NAME="E7:22_7_1_3"/><i><font color="Green" title="E32">A80</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_5.html#T2" title="YELLOW_5:th.2">YELLOW_5:2</a></i>;<br/><a NAME="E8:22_7_1_3"/>
<font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">a</font>
 ;<br/><a NAME="E9:22_7_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:22"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_5.html#T6" title="YELLOW_5:th.6">YELLOW_5:6</a></i>;<br/><a NAME="E11:22_7_1_3"/><b>then </b><i><font color="Green" title="E33">A81</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E3:22_7_1_3"><i><font color="Green" title="E29">A77</font></i></a>, <a class="txt" href="lattice8.html#E4:22_7_1_3"><i><font color="Green" title="E30">A78</font></i></a>, <a class="txt" href="lattice8.html#E6:22_7_1_3"><i><font color="Green" title="E31">A79</font></i></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><a NAME="E13:22_7_1_3"/><b>then </b><i><font color="Green" title="E34">A82</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E7:22_7_1_3"><i><font color="Green" title="E32">A80</font></i></a>, <a class="txt" href="lattice8.html#E11:22_7_1_3"><i><font color="Green" title="E33">A81</font></i></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><a NAME="E14:22_7_1_3"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:22"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="lattice5.html#D6" title="LATTICE5:def.6">LATTICE5:def 6</a></i>;<br/><a NAME="E15:22_7_1_3"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:22"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T7" title="YELLOW_5:th.7">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c16">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E13:22_7_1_3"><i><font color="Green" title="E34">A82</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_7_1_3"><i><font color="Green" title="E28">A76</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></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" title="E28">A83</font></i>: 
( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:22_7_1_4"><i><font color="Green" title="E28">A83</font></i></a>, <a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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" title="E27">A84</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">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" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E1:22_8"/><i><font color="Green" title="E28">A85</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">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" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E1:22_8"><i><font color="Green" title="E28">A85</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">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" title="E29">A86</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:22_8_1_1"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_8_1_1"><i><font color="Green" title="E29">A86</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></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" title="E29">A87</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E1:22_8_1_2"><i><font color="Green" title="E29">A87</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_8_1_2"><i><font color="Green" title="E29">A87</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></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" title="E29">A88</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:22_8_1_3"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_8_1_3"><i><font color="Green" title="E29">A88</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></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" title="E29">A89</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E1:22_8_1_4"><i><font color="Green" title="E29">A89</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_8_1_4"><i><font color="Green" title="E29">A89</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></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" title="E29">A90</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E1:22_8_1_5"><i><font color="Green" title="E29">A90</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:22_8_1_5"><i><font color="Green" title="E29">A90</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>
;<br/><b>hence </b><a NAME="E3:22_8_1_5"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_8_1_5"><i><font color="Green" title="E29">A90</font></i></a></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" title="E29">A91</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:22_8_1_6"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_8_1_6"><i><font color="Green" title="E29">A91</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></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" title="E29">A92</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E1:22_8_1_7"><i><font color="Green" title="E29">A92</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_8_1_7"><i><font color="Green" title="E29">A92</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></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" title="E29">A93</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:22_8_1_8"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:22_8_1_8"><i><font color="Green" title="E29">A93</font></i></a>, <a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></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" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font> holds  <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">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" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">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" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> ) )
 </span><b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E7:22"><i><font color="Green" title="E20">A5</font></i></a></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" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E8:22"><i><font color="Green" title="E21">A8</font></i></a></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" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E9:22"><i><font color="Green" title="E22">A18</font></i></a></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" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E10:22"><i><font color="Green" title="E23">A30</font></i></a></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" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E11:22"><i><font color="Green" title="E24">A48</font></i></a></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" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E12:22"><i><font color="Green" title="E25">A56</font></i></a></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" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E13:22"><i><font color="Green" title="E26">A66</font></i></a></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" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">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" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E14:22"><i><font color="Green" title="E27">A84</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E16:22"/>
 <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>
 <b>by </b><i><a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a></i>;<br/>


</div>
